NEW QUANTUM ALGORITHMS AND ANALYSES FOR HAMILTONIAN SIMULATION

By

Jacob Watkins

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy

2024

ABSTRACT

Digital quantum Hamiltonian simulation is, by now, a relatively mature field of study; however,

new investigations are justified by the importance of quantum simulation for scientific and soci-

etal applications. In this dissertation, we discuss several advances in circuit-based Hamiltonian

simulation.

First, following two introductory chapters, we consider the mitigation of Trotter errors using

Chebyshev interpolation, a standard yet powerful function approximation technique. Implications

for estimating time-evolved expectation values are discussed, and a rigorous analysis of errors and

complexity show near optimal estimation of dynamical expectation values using only Trotter and

constant overhead. We supplement our theoretical findings with numerical demonstrations on a 1D

random Heisenberg model.

Next, we introduce a computational reduction from time dependent to time independent Hamil-

tonian simulation based on the standard (𝑡, 𝑡′) technique. Our approach achieves two advances.

First, we provide an algorithm for simulating time dependent Hamiltonians using qubitization, an

optimal algorithm that cannot handle time-ordering directly. Second, we provide an algorithm for

time dependent simulation using a natural generalization of multiproduct formulas, achieving higher

accuracies than product formulas while retaining commutator scaling. Rigorous performance anal-

yses are performed for both algorithms, and simple numerics demonstrate the effectiveness of the

multiproduct formulas procedure at reducing Trotter error.

Finally, we consider several practical methods for near-term quantum simulation. First, we

consider the analog quantum simulation of bound systems with discrete scale invariance using

trapped-ion systems, with applications to Efimov physics. Next, we discuss the Projected Cooling

Algorithm, a method for preparing bound states of non-relativistic quantum systems with localized

interactions based on the dispersion of unbound states. Lastly, we discuss the Rodeo Algorithm,

a probabilistic, iterative, phase-estimation-like protocol which is resource-frugal and effective at

measuring and preparing eigenstates. Concluding remarks and possible future directions of research

are given in a brief final chapter.

Dedicated to my late grandmother, Elizabeth,
who swore to be the first to call me "Dr. Watkins."

(I told her my committee would beat her to it.
She insisted otherwise.)

iii

ACKNOWLEDGEMENTS

This work is the culmination of six years of graduate study in physics and quantum computing,

which itself builds upon a long journey of educational and personal development. I could not have

done this without the many people and institutions that have supported me along the way.

First, I thank the Earth for supporting lifeforms who do silly things like study quantum com-

puting. May these pages not be a waste of your resources.

I thank my family, who I’ve missed dearly, for their encouragement and support. In particular:

my mother Diana, for setting an example of good character, for being tender, and for knowing how

to handle a son who was hard on himself; my grandfather Rob, for contributing to my love for the

outdoors and showing me how to cut the Thanksgiving turkey; my Aunt Dawn, for being the only

one who enjoys board games and karaoke as much as I do; my Uncle Gene, for teaching me to

like the taste of oyster and pause to appreciate the flowers; and my cousins Ethel and Elizabeth, for

being like siblings to me.

Friends, past and present, give warmth and meaning to my life. I thank my "camp friends" (you

know who you are), and Camp Orkila itself for letting me sing silly songs and instilling values I

retain today. I thank friends I made at the University of Washington, especially Brian, Frances,

Katrina, and Mark. My visits to you, and yours to me, have been a source of strength during my

PhD. Thanks to Keigan and Elliott for uplifting phone calls and for bringing Rosemary into this

world. To Kyndra and Josh; I will always appreciate our adventures into the woods.

A long list of mentors and educators have guided me since deep into my childhood. I thank

my Taekwondo instructor Eric Shields for teaching me about perseverance and indomitable spirit.

I thank the wonderful educators of Shaw, Kalles, and Puyallup High who have given me a well-

rounded liberal arts education. Particular thanks go to Ms. Kreiger, for calling my mother to tell

her I did well on an assignment, and to Ms. Kooser, for an engaging, hands-on social studies

curriculum. Thanks to Mr. Ryan, for facilitating most of my social life via the PHS band program.

Of course, a heartfelt thanks to Mr. Segers, my high school physics teacher, for tolerating my

(frequent) class-entry skits and my (occasional) anxieties about grades.

iv

More recently, thanks to Ben Hall, my first office mate and friend in graduate school, for

showing me how cool quantum computing is; Maria Violaris for singing Taylor Swift parodies with

me at the premier quantum information conference and showing me that "punting" meant more

than kicking an American football. Thanks to Kirtimaan Mohan, Katie Hinko, Nick Ivanov, Rachel

Barnard, Matt Oney and others who have helped me explore my interests in education while being

supportive mentors and collaborators.

Despite being housed in a major nuclear science lab for most of my studies, the reader may

notice that my research has strayed quite far from nuclear physics proper. I thank Dean Lee for

giving me ample freedom and encouragement to explore my interests. Thanks as well to Nathan

Wiebe for hosting me for a summer in Toronto, and for mentoring me in the kind of rigorous

quantum algorithms research exhibited in much of this thesis. I also thank Ale Roggero for being

a supportive and patient mentor and collaborator, and for being there for a young graduate student

struggling through his first real research talk.

Finally, I thank my partner, Brita, for her excellent pies and shakshuka, introducing me to Mar-

garet Atwood, and converting me to coffee. We have supported each other through the difficulties

of the pandemic and come out stronger people. I love you like a rock.

v

CHAPTER 1

HOUSEKEEPING . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

TABLE OF CONTENTS

CHAPTER 2

BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
.
3
2.1 Quantum Mechanics and Challenges to Realism . . . . . . . . . . . . . . . . .
6
2.2 Computer Science and the Role of Physics . . . . . . . . . . . . . . . . . . . .
8
2.3 Quantum Computing, an Overview . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The Case for Quantum-Based Quantum Simulation . . . . . . . . . . . . . . . 12
2.5 Quantum Hamiltonians: A Closer Look . . . . . . . . . . . . . . . . . . . . . 16
2.6 Classical Simulation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Quantum Simulation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Mathematical Reference

CHAPTER 3

TROTTER ERROR MITIGATION . . . . . . . . . . . . . . . . . . . . 36
3.1
Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Stability Analysis .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 The Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Application to Dynamical Observables . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Numerical Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
. 63
3.7 Discussion . .
. 65
. .
3.8 Proofs . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .
. .

.
.

.
.

.
.

.

.

CHAPTER 4

TIME DEPENDENT HAMILTONIAN SIMULATION THROUGH
DISCRETE CLOCK CONSTRUCTIONS . . . . . . . . . . . . . . . . . 71
Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
. 93
. 102

4.1
4.2 The Clock Space
.
4.3 Finite Clock Spaces .
4.4 Time Dependent Qubitization . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Discussion .

.

.

.

.

.

CHAPTER 5

MULTIPRODUCT FORMULAS FOR TIME DEPENDENT
. 103
SIMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Definition and Effectiveness
5.3 Time Dependent Multiproduct Simulation . . . . . . . . . . . . . . . . . . . . 111
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Error Analysis . .
. .
5.5 Time Step Analysis .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
.
5.6 Query Complexity .
5.7 Numerical Demonstrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.8 Discussion .
. 143
5.9 Algorithm for Time Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

CHAPTER 6

A SIMULATION MIXED BAG . . . . . . . . . . . . . . . . . . . . . . 146
6.1 Discrete Scale Invariance on Trapped-Ion Systems . . . . . . . . . . . . . . . . 146

vi

6.2 Projected Cooling Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
. 159
6.3 Rodeo Algorithm .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

CHAPTER 7

CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . 173

BIBLIOGRAPHY . .

. .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

vii

CHAPTER 1

HOUSEKEEPING

This dissertation is concerned primarily with the task of simulating a Hamiltonian on a quantum

computer. What is a Hamiltonian? Why do we want to simulate it? What is a quantum computer?

Those already in-the-know, who want to skip to the technical advances of this thesis, are

welcome to survey the chapters to find what they are looking for. I expect they are already inclined

to do so! Those who want a little more context for this thesis, and to learn some of the big ideas

underlying the field, are encouraged to continue on to the next chapter. There I will provide a

sweeping, but necessarily brief, survey of the big ideas in the field of digital quantum Hamiltonian

simulation, steadily working towards the more technical advances of this work.

The Abstract of this dissertation provides, as expected, a synopsis of all topics covered. All

chapters are essentially independent of each other, with the partial exception of Chapter 5, which

relies some ideas from Chapter 4 to support a conjecture. Moreover, all but Chapter 6 correspond to

a single, self-contained research project. The penultimate Chapter 6 covers a mixed bag of projects

to which I contributed.

All of the projects discussed in this thesis have corresponding publications or preprints, and

references to these works are provided near the beginning of the relevant chapter or section. When

deciding what to include here, and what to leave to those works, I use several criteria. First,

since this thesis serves as a compendium of my work, I focused chiefly on my contributions to a

given project. Sometimes though, for the sake of a self-contained document, I include results and

derivations that are primarily due to my collaborators. I will indicate clearly when this is the case.

Finally, for various reasons, there are results from these projects that did not make their way into the

corresponding publication, especially in the miscellaneous Chapter 6. By including those here, I

hope to complement the publications by supporting and extending their findings through numerics

or derivations.

The flavor of this thesis is analytical, in at least two senses. First, in the mathematical sense. To

approach a problem "analytically" means to utilize tools of mathematical proof and derivation, in

1

contrast to numerical calculation. The central results are proofs and analytical bounds on error and

computational complexity. Numerics, however, are used to provide assurance and to see the "actual"

performance in a way that complexities cannot showcase. These benchmarks are usually far from

complete, suggesting an obvious path for additional research. Second, this thesis is analytical in

that it is primarily concerned with analyzing something, namely quantum algorithms. Although we

propose novel simulation methods, they are typically variations on existing tools. The performance

and errors analyses are likely the major technical advancement of this dissertation. I believe that

such careful analyses provide firm guideposts for those who wish to apply algorithms to specific use

cases. Hopefully, our methods for estimating algorithmic resources can be useful for the analysis

of quantum algorithms developed in the future.

Often in practice, analytical error bounds fall short of representing the typical error of a

simulation method [25, 66]. This is mostly good news, meaning performance is often much better

than expected. What, then, is the value of such bounds if they fail to capture the "actual" behavior

of the method? Worst-case error and resource bounds represent a first important step towards

understanding the behavior and capability of a method, providing us the most robust guarantees

of how well an algorithm will perform. This is only part of the picture, and while numerical

experiments can provide more insight, there is additional work for theorists as well. For example,

recent work on average-case hardness for Trotter simulations likely represents a step towards a

fuller understanding of the "typical" hardness [23].

Without further ado, please enjoy what this dissertation has to offer.

I hope you find these

chapters helpful not only for technical content, but for inspiration and ideas for your own pursuits.

2

CHAPTER 2

BACKGROUND

I expect there will be as many readers of this thesis who are newcomers to quantum computing as

those who are quantum algorithms experts. Thus, I am motivated to dedicate a chapter to provide

both background and inspiration for the technical results that follow. We will start with broad scope

and little detail, gradually narrowing our focus to the new stuff. More detailed background on

specific research is given at the beginning of each chapter.

Quantum information science, which includes quantum computing, is a relatively young dis-

cipline which overlaps several technical fields, particularly physics and computer science. My

approach to this chapter is to discuss each of these domains separately, then their surprising inter-

connection made most obvious by (but not reliant on) quantum computing. With these ingredients

in place, we then introduce Hamiltonian simulation, which relates to the computation of either

closed, naturally occurring quantum systems or problems with equivalent mathematical structure.

I hope the reader finds these short surveys valuable to understand the more particular and technical

work of later chapters.

2.1 Quantum Mechanics and Challenges to Realism

The developments in physics which began at the turn of the 20th century, were, in many respects,

parallel with those found in the arts in that same period. As the modernists eschewed accurate

portrayals for abstract figures and geometries, the physicists grappled with phenomena increasingly

removed from regular experience. And, like the modernists, these new ways of doing physics

were met with some backlash. Despite this, the resulting theories, namely relativity and quantum

mechanics, were better at explaining the world around them than the earlier "classical" physics. Yet

their character was so strange that it led some physicists, notably Dirac1, to emphasize mathematics

over the senses in formulating physical theories.

This is especially true for quantum mechanics, which on the surface said a number of very

1"I learned to distrust all physical concepts as the basis for a theory.

Instead one should put one’s trust in a

mathematical scheme, even if the scheme does not appear at first sight to be connected with physics." [92]

3

strange things. Particles were neither here nor there until measured, the story goes, seemingly

defying the scientific tenet of realism. Particles may be waves, waves may be particles. More

truthful than such common refrains is that quantum mechanics provides a relatively well-defined

framework for accurate calculations of particles, atoms, molecules, and nuclei, even as the theory

appeared unintuitive or even nonsensical. The mathematics of quantum mechanics nicely captured

a variety of phenomena which eluded classical treatment, but the classical theory made more sense.

Yet even today, the meaning of quantum mechanics remains mainly unresolved. In response to this

absurdity, the prevailing attitude amongst quantum physics practitioners is captured in the pithy

mandate: "Shut up and calculate." The meaning: don’t worry about what the theory means, per se,

just worry about what it predicts. While it’s easy to criticize this point of view, delaying thorny

questions of interpretation arguably allowed for more rapid understanding of physical phenomena

in the decades following the invention of quantum mechanics.

One unsettling aspect of quantum mechanics is its intrinsic nondeterminism. The theory

only predicts probabilities of certain outcomes in a physical experiment, where "experiment"

is interpreted broadly as any means by which observers (such as people) experience the world

around them. While probabilities had appeared earlier in statistical mechanics and its connection

to thermodynamic entropy, their appearance here in a fundamental physical theory was a notable

break from the past, and carried unsettling philosophical implications. It led Einstein, Podolsky,

and Rosen to argue that quantum mechanics was actually an incomplete theory of reality [42], and

a more complete understanding, even if impossible to achieve by mere mortals, would reveal an

underlying determinism. Some alternate theories, most notably Bohmian mechanics [55], purport

to restore determinism to quantum by relegating all chance to inaccessible knowledge of the particle

trajectories. It was shown in a groundbreaking work by John Bell that certain reasonable "hidden

variable theories" make predictions distinct from those of quantum mechanics [12]. Experiments

on the matter came out in favor of quantum theory [7, 49], and these contributions were rewarded

with the 2022 Nobel Prize in Physics. The bottom line is that it is hard to restore determinism,

or even conventional probability, to quantum mechanics without violating other cherished physical

4

principles such as locality. In contrast to hidden-variable theories, which seem to deny quantum

mechanics as it is, the many worlds interpretation [38] asks us to take quantum theory at face value,

including the reality of the quantum wavefunction as a description of all phenomena.

So far, I have emphasized that quantum physics is a radical departure, conceptually, from

classical physics. Yet this belies the fact that, in formulating quantum mechanical models, classical

models are often used as a starting points. It is easiest to start with existing tools when trying to

create new ones. The process of taking a classical theory and tweaking it to describe a quantum

system is known as "quantization." It turns out that not all ideas from classical physics are equally

suitable for quantization. For example, the Newtonian framework, in which changes in motion

are generated by forces, does not have a great correspondence to quantum mechanics principles.

Rather, the most natural jumping off point for quantum mechanics is the Hamiltonian formulation,

named after Irish mathematician, astronomer, and physicist William Rowan Hamilton.

In this

framework, a classical system has physical configurations given by a number of coordinates 𝑞.

For example, the location and orientation of an airplane may be exactly represented by a set of 6

numbers. Each coordinate 𝑞 has a corresponding conjugate momentum 𝑝, which in simple cases

may be seen as expressing a "velocity" for 𝑞. Specifying all coordinates 𝑞 and momenta 𝑝 gives

a complete specification of the system in the sense that any "observable quantity" 𝑂 is a function

𝑂 (𝑞, 𝑝) of the coordinates and momenta.

One uniquely special observable is the Hamiltonian 𝐻 (𝑞, 𝑝), which provides the total energy

of the system as a function of its coordinates and momenta. It also contains all information about

future states of the physical system. That is, the classical Hamiltonian defines a set of differential

equations

𝑑𝑞
𝑑𝑡

=

𝜕𝐻
𝜕 𝑝

,

𝑑𝑝
𝑑𝑡

= −

𝜕𝐻
𝜕𝑞

(2.1)

which, when solved, provide the state of the system at any subsequent time. More succinctly, 𝐻

encodes the dynamics of the physical system in question. The importance to physics is immediate.

One of the primary goals of physics is to understand a phenomena well enough to make future

predictions given current data. Prediction is more powerful, and impressive, than retroactive

5

explanation. The Hamiltonian provides all the information needed to make these predictions.

In an exactly analogous manner, the quantum Hamiltonian 𝐻 (we use the same symbol) encodes

all of the dynamics of closed quantum system. Given a system described by wavefunction |𝜓⟩ the

dynamics are found by solving the famous Schrödinger equation.

𝑖𝜕𝑡 |𝜓𝑡⟩ = 𝐻 |𝜓𝑡⟩

(2.2)

Formally, 𝐻 in the quantum setting is a Hermitian operator on a Hilbert space, and |𝜓𝑡⟩ is a vector-

valued function on this space. The main point is that solving (2.2) is a fundamentally important

task for understanding the dynamics of physical phenomena. Solving this equation allows for

understanding the formation of the elements, the properties of molecules and materials, and the

fundamental constituents of nature.

In practice, solving (2.2) is far too difficult with even today’s best computational devices and

cleverest tricks. Instead, people come up with a number of clever partial solutions and approximation

schemes, to various degrees of success. In talking about what can be computed efficiently and what

cannot, however, we have already begun to enter a different domain worth discussion: computer

science.

2.2 Computer Science and the Role of Physics

In Scott Aaronson’s Quantum Computing Since Democritus, he writes that computer science

"is a bit of a misnomer." Rather than being about computers, in the particular sense of desktops,

servers, and smart phones, he views it as "the study of the capacity of finite being such as us to learn

mathematical truths." [2] Aaronson understands that "mathematical truths" encompasses more than

what is sought by professional mathematicians. It could involve finding the shortest route to work

(Dijkstra’s Algorithm), or predicting protein structure from an amino acid sequence [74]. Such

tasks, in light of modern science, are likely viewed as being intrinsically mathematical and thus

within the domain of computation. However, with recent developments in artificial intelligence,

particularly Large Language Models (e.g., chatGPT), even more domains of human activity have

been made amenable to computational treatment.

6

The diversity of computational problems is paralleled by the diversity of entities which can serve

as a computational medium. Indeed, prior to the development of general-purpose, digital computers

in the mid-20th century (and even following that), the term referred to human computers who

performed much of the calculations for scientific, industrial, and governmental applications (see,

e.g., [117]). While standard, modern computers are electronic, computers could in principle be

made from billiard balls [48] or even water [3]. In some sense, however, all of these computers

are equivalent to a Turing machine, an idealized computer consisting of a tape for manipulating

symbols, an input program, and several internal states. One way to express this equivalence is that

a Turing machine may be used to simulate the calculations performed by any of these other models.

The hypothesis that "computable by Turing machine" captures the notion of what is computable

is referred to as the Church-Turing thesis [32]. No serious challenge to this thesis has been sustained

in the near-century since it was proposed. If the thesis holds, it seems to suggest that details about

the physical system performing the computation may be abstracted away.

If so, one shouldn’t

expect questions in physics to have much bearing on computer science, besides the practicalities of

engineering an effective computational device.

Although understanding computability, i.e. what problems may be solved by computation, is

important, we are left without an understanding of which problems are "practically" solvable on real

world computers. For example, let’s return to the Schrödinger equation (2.2). This equation can

be solved straightforwardly, to arbitrary accuracy, given enough time, space, and energy. However,

the amount of these resources needed is prohibitively large for interesting instances. No one wants

to wait 1000 years for a single result. Questions of what may be computed efficiently falls under

the purview of the subfield of computational complexity. Nowadays, most aspects of theoretical

computer science are concerned with complexity, and computability is a relatively closed subject.

If we are actually interested in computational complexity, rather than computability, it makes

sense to consider a modified Church-Turing thesis which deals more with the former than the latter.

In particular, we may ask: Is "efficiently computable by Turing machine" equivalent to "efficiently

computable by actual, physical computers"? The so-called strong Church-Turing thesis is the

7

assertion that this is the case, and because of the new word "efficiently" in the above question, the

claim is indeed stronger than what either Church or Turing originally proposed. Though the claim

is stronger, the evidence for its truth is correspondingly weaker. In fact, existing evidence suggests

that quantum computers, if constructed, could solve problems efficiently that Turing machines could

not [118, 120]. The potential for quantum computers to solve problems relevant to society has driven

major investment from government and industry [53]. From a more fundamental perspective, the

power of quantum computing suggests a greater interplay between physics and computer science

than has been historically explored. Are there other physically realizable models of computation

even more powerful than classical or quantum computers? In this direction, work by Aaronson

has shown how computers based on hidden variable theories would be slightly more powerful than

standard quantum computers [1]. The upshot of these developments is that physics seems to play

an essential role in a fundamental computer science question: what computational tasks may be

efficiently performed?

2.3 Quantum Computing, an Overview

In the last section, we rapidly converged on the notion of a quantum computer, and here we

discuss in more detail what this means. As we approach our primary topic, Hamiltonian simulation

on a quantum computer, I will use increasingly precise and technical language, and no longer avoid

mathematics. Readers with background in linear algebra and complex numbers are encouraged to

consult standard resources for more thorough introductions to quantum computing [101].

At a high level, a quantum computer is nothing more than a computer based on the laws of

quantum mechanics. Any computation requires, abstractly, the encoding of information and its

manipulation by certain operations to achieve a result. For example, a classical computer may

perform addition by storing two numbers in binary registers, then manipulating these registers in a

specified way to get the sum on one of the two registers. A quantum computer, by contrast, stores

its information as quantum states, and manipulates these states. As a caution, although any laptop is

describable, in principle, in quantum mechanical terms, the way they store information and perform

operations is most aptly described as "classical." This is true of essentially any computational device,

8

save the handful of quantum computers under development today.

How do we model the workings of a quantum computer concretely? Just like with clas-

sical computers, many possible computational models exist. Quantum Turing machines [37],

measurement-based [111], and adiabatic quantum computation [4] are several well-explored ex-

amples. But the most popular approach by far is the circuit-based model of quantum computation,

which we will now explain in detail. The reader will benefit in having some background in the

classical circuit model of computation, or experience with real digital logic circuits.

Figure 2.1 provides an example of a quantum circuit. As with the classical circuit model,

quantum circuits have wires and gates that feed forward (no feedback loops), but here the wires

contain quantum information. Instead of well-defined bits in the 0 or 1 state, wires carry quantum

bits, or qubits. In isolation, a qubit can have a state in the form

𝑐0|0⟩ + 𝑐1|1⟩.

(2.3)

Here, the symbols |0⟩ and |1⟩ are "kets" which take on a precise meaning as orthonormal vectors in

a two-dimensional complex inner product space, but can be thought of informally as the "definite"

states that the qubit can take. The coefficients 𝑐𝑖 are complex numbers that are often called

"amplitudes." The fact that 𝑐𝑖 does not have to be a positive real number is the crucial difference

between quantum computing and probabilistic, a.k.a. Monte Carlo, computing. The probability of

measuring 0 or 1 is given by |𝑐0|2 and |𝑐1|2, respectively, but before a measurement is performed

the amplitudes can exhibit interference. In order to have total probability one, we must have the

following normalization condition.

|𝑐0|2 + |𝑐1|2 = 1

(2.4)

For vectors |𝜓⟩ and |𝜙⟩, their inner product is denoted ⟨𝜙|𝜓⟩. In this language, the normalization

condition for a qubit in state |𝜓⟩ can be written as ⟨𝜓|𝜓⟩ = 1. In this thesis, the term state vector

will mean the normalized vector used to mathematically represent the state of our quantum system.

Many readers will be familiar with the more general density matrix representation of quantum

states, but because of our focus on closed-system dynamics we will not have much need for this

more complicated formalism.

9

𝑋

𝐻

𝑇

𝑆

Figure 2.1 Example of a quantum circuit on three qubits. Information is carried on the wires as a
collective quantum state, which is a superposition of possible values of the bitstrings on the
register. This information is manipulated by gates which act on one or more of the qubits. Partial
information about the quantum state is obtained by measurements, here represented by meters.
The measurements also affect the state in accordance with quantum mechanics.

For any interesting computation, combining multiple qubits together will be necessary. Such a

collection will be referred to as a quantum register. As with combining any two (distinguishable)

quantum mechanical systems, joint qubit states are described formally through the tensor product

of the individual state spaces. Any state vector |Ψ⟩ on the joint system is a linear combination of

product vectors of the form

|𝜙⟩ ⊗ | 𝜒⟩

(2.5)

where |𝜙⟩ and | 𝜒⟩ are state vectors on each individual space. As an important example, a state

vector on a collection of 𝑛 qubits may be generally expressed as a sum

|𝜓⟩ =

∑︁

𝑐𝑏 |𝑏⟩

𝑏∈{0,1}𝑛

(2.6)

over all bitstrings 𝑏 = 𝑏1𝑏2 . . . 𝑏𝑛. There is an underlying tensor product |𝑏⟩ ≡ |𝑏1⟩ ⊗ |𝑏2⟩ ⊗ · · · ⊗

|𝑏𝑛⟩ ≡ |𝑏1𝑏2 . . . 𝑏𝑛⟩ that is often convenient to leave implicit. Whenever the joint state |Ψ⟩ cannot

be written as a product vector, it is said to be entangled. Generically speaking, almost all quantum

states are entangled, in the sense that choosing a random state has vanishingly small probability

of being a product state for systems with more than a small number of states [144]. Entanglement

is a necessary condition for quantum computers to exhibit superior performance over classical

computers, though identifying the source of "power" quantum computation is a somewhat subtle

issue [69].

To summarize, the objects of our quantum computer are qubits, and their collective state is

10

given by quantum state vector, i.e. a normalized vector in the tensor product over each qubit vector
space. More concisely, this is just a normalized vector in C2𝑛

for 𝑛 qubits. Measuring the qubits

returns a bitstring with probability given by the squared amplitude.

We must now establish the appropriate operations for our quantum computer. Naturally, our

operations should not take us outside the set of allowed states, namely the quantum state vectors

described above. Moreover, empirical evidence suggests that quantum mechanical operations are

all linear, so that operations on the full state may be understood by considering the operations on

each component of the superposition. We are therefore led to consider gates as unitary operations

acting on a subset of qubits. Unitary operations can be defined in a number of equivalent ways, all

of which relate to the idea of preserving the norm of the state vector. More precisely, a unitary 𝑈

is a linear operator on an inner product space such that

∥𝑈 |𝑣⟩∥ = ∥ |𝑣⟩ ∥

(2.7)

for any vector 𝑣 in the space, where ∥ · ∥ represents the Euclidean, or 𝐿2, norm. A quantum gate, or

simply "gate" in this context, is a unitary operation acting on a small number of qubits in a circuit.

We generally represent gates as boxes with wires passing through.

Today, computer programmers rarely work at the level of digital logic and gates on their laptop.

Instead, programmers work at higher levels of abstraction to skirt minute details, accomplishing

more as a result. In this thesis, we will seek to analyze quantum algorithms, and it will be easier

and more insightful if we consider larger chunks, or subroutines, used to carry out the method.

In the analysis, we will call these subroutines oracles: "black boxes" that are used as part of the

algorithm, whose inner workings we either don’t know or delay considering. For the quantum

context, the oracle 𝑂 will be a unitary operation on some number of qubits. As an example of an

oracle, consider a function 𝑓 from 𝑚-bit to 𝑛-bit strings. From this classical operation, we can

define a unitary oracle 𝑈 𝑓 that computes 𝑓 . Given two quantum registers of length 𝑛 and 𝑚 in state

|𝑥⟩ ⊗ |𝑎⟩ ≡ |𝑥⟩ |𝑎⟩, define 𝑈 𝑓 as

𝑈 𝑓 |𝑥⟩ |𝑎⟩ = |𝑥⟩ | 𝑓 (𝑥) ⊕ 𝑎⟩ ,

(2.8)

11

where ⊕ is addition modulo 2, and extend the definition to general vectors by linearity. It can be

checked that 𝑈 𝑓 is unitary, and when 𝑎 = 0 it is clear that 𝑈 𝑓 computes 𝑓 (𝑥). Such oracles are

used, for example, in Shor’s order-finding algorithm, where 𝑓 (𝑥) = 𝑎 · 𝑥 is multiplication by some

integer 𝑎. In the context of this dissertation, we will define oracles that compute the parameters

of a Hamiltonian or evolve a quantum register according to some Hamiltonian. Oracles may also

encode specific observables to be measured during a quantum algorithm. Without knowing the

computational cost of implementing an oracle 𝑂, it is impossible to know the cost of any algorithm

utilizing 𝑂 as a subroutine. One of the benefits of using oracles is to break down the problem into

smaller pieces: first implementing the oracle, then implementing the algorithm given the oracle.

Another benefit is abstraction: we can analyze the general oracle problem, then apply those results

to any particular instance of that oracle. In the example above, this might entail different functions

𝑓 , some of which are easy to compute, others of which are uncomputable!

2.4 The Case for Quantum-Based Quantum Simulation

The advent of quantum mechanics carried great promise to better understand a range of physical

phenomena, but challenges remained that were more about computational feasibility than theoretical

understanding. As expressed by Paul Dirac [39] regarding the invention of quantum mechanics,

The underlying physical laws necessary for the mathematical theory of a large part of

physics and the whole of chemistry are thus completely known, and the difficulty is

only that the exact application of these laws leads to equations much too complicated

to be soluble.

Though there appears "only" one barrier to solving all of chemistry, it is indeed a very large one.

Massive supercomputing resources are needed to solve, starting from quantum mechanics, even

relatively small molecules. As a result, heuristic and phenomenological techniques are employed

in practice, which are less expensive but also less reliable and general. Examples of computational

techniques for quantum many-body problems include a slew of variational techniques, such as

Hartree-Fock and Coupled Cluster, and dynamical methods such as Molecular Dynamics [33].

12

Despite these clever and well-established approaches, we emphasize that there remains a lack of

general, efficient methods for classical simulation of quantum dynamics.

Although efficient quantum simulation would likely not obviate the need for high-level, domain-

specific concepts in chemistry, materials science, etc., such simulations would nevertheless be

valuable to science and human knowledge. Unfortunately, it is believed that classical computers

could never simulate quantum physics efficiently, meaning they couldn’t solve the Schrödinger

equation (2.2) in interesting instances without intractable amounts of resources. After a hundred

years of effort, effective classical methods for computing general quantum properties have simply

not been found, although discovering such methods would be both useful and philosophically

profound.

Some of the earliest explorations into quantum computing were motivated by the desire for

efficient quantum simulation [46, 37]. As it happens, not only can quantum computers simulate

physical Hamiltonians efficiently, as we shall see in Section 2.7, but also this task fully captures the

power of quantum computing. More precisely, it turns out that 𝑘-local Hamiltonian simulation is

BQP-complete. Here 𝑘-local refers to Hamiltonians of the form

Γ
∑︁

𝐻𝛾

𝐻 =

(2.9)

𝛾=1
where each 𝐻𝛾 only operates on at most 𝑘 qubits. In words, interactions across the system are built

up from interactions involving only a small number of constituents. BQP is, loosely speaking, the

class of problems which may be efficiently solved by a quantum computer. "BQP-complete" refers

to the fact that 𝑘-local Hamiltonian simulation is both in BQP (quantum computers can do it) and

also that any problem in BQP may be encoded as a 𝑘-local simulation problem (in fact, 2-local)2.

Thus, we may view digital quantum circuits and local Hamiltonian simulations as two sides of

the same coin. This may not come as a surprise to many physicists, who are used to viewing all

quantum operations as arising from an underlying Hamiltonian. The importance is that, if quantum

computers can do anything interesting at all compared to classical computers, then Hamiltonian

simulation should be one of those things.

2For experts: this encoding may require at most polynomial overhead in spatial and temporal resources.

13

A heuristic, imperfect argument that quantum computers can simulate quantum systems ef-

fectively is that they also are quantum mechanical. By having the computer mimic the quantum

transformations occurring in the physical system, one might hope to achieve what the classical

computer cannot replicate efficiently. This argument provides some insight but also, by itself, is

unconvincing. What is the essential difference between quantum simulation and simply "watching"

a quantum system of interest "do its thing"? There are, in fact, major differences:

1. You may not have much access to the system of interest. For example, it is challenging,

expensive, or impossible to send a probe to the sun’s core to gain information on nuclear

processes. A protein under study may behave differently in a test tube than in situ. Essentially,

there are many phenomena that are hard to measure directly under the desired circumstances.

2. There may be very little control over the parameters of the system of interest, or little ability to

measure a wide variety of properties. Digital quantum simulation gives an enormous (though

not limitless) degree of control over the model parameters and read out of the desired results.

Identical trials of the same simulation could, in principle, be prepared as desired. The ability

to tweak aspects of the simulation leads to a greater understanding of the phenomenon under

study. By contrast, barring a highly tunable experimental setup, a system "just is" and doesn’t

necessarily provide insight.

In short, simulation is more than just experiment, both classically and quantumly. In Section 2.7,

we will see how Hamiltonian simulation algorithms can be quite abstracted from any particular

information about the system, thus earning designations such as "computation" and "algorithm".

The classical intractability of quantum simulation is a more subtle issue than is often let on

in popular, or even research, presentations, and we take a moment to challenge these oversimpli-

fications. The common argument starts and ends with the exponential growth in the number of

elementary states as the quantum system size increases. Recall the expression (2.6) for an arbitrary

𝑛-qubit state. Without additional structure, it appears the state |𝜓⟩ requires 2𝑛 complex numbers to

describe, one for each bitstring 𝑏. Each additional qubit doubles the number of states, and 2100 is

14

already an enormous number for just 100 particles. Normalization conditions do not help much,

nor does the irrelevance of global phase. This amount of data is inefficient to store in memory, let

alone perform operations on.

The problem with the above argument is that any physical system, quantum or not, has such a

scaling: a system of 𝑛 particles with 𝑑 single-particle states has a total of 𝑑𝑛 states. Why is there no

objections to exponentiality in this context? To explore this, we look at standard probability theory,

which is the closest jumping-off point for quantum mechanics. We may express the state | 𝑝⟩ of 𝑛

probabilistic bits3 in the following suggestive form.

| 𝑝⟩ =

∑︁

𝑝𝑏 |𝑏⟩

𝑏∈{0,1}𝑛

(2.10)

Here the collection of 2𝑛 𝑝𝑏’s forms a probability distribution, where 𝑝𝑏 is the probability of

observing bitstring 𝑏. Compare with equation (2.6), and we seem to be facing the same conundrum.

And indeed we are, if the goal is to represent the full distribution | 𝑝⟩. However, in applications,

we would usually rather sample from the distribution rather than express it. The difference is the

same as flipping a fair coin vs. writing a list (1/2, 1/2). Sampling typically requires much less

spatial overhead, and can be performed using probabilistic bits and operations. The situation in

quantum computing is very, very similar. We cannot "see" the output state |𝜓⟩ in entirety, but

measure and achieve some result with some probability. Thus, the source of quantum computing’s

power appears to come not just from an exponential number of amplitudes, but also the way these

amplitudes interfere under unitary operations.

To summarize, although we have a comprehensive theory of quantum phenomena, it is currently

very difficult to compute the consequences of this theory. Further, we should not believe generic,

efficient classical methods will ever be found. On the other hand, the use of quantum computers

for quantum simulation is an intuitive and promising solution to this challenge. We shall see in

Section 2.7 that there are several efficient quantum algorithms for Hamiltonian simulation. For

the most part, the only remaining obstacle is a very difficult engineering problem: building better

3Note that the notation | 𝑝⟩ is not suggestive of anything quantum. I use it here in this context to emphasize that

the ket is just notation.

15

quantum computers4. Building effective quantum hardware is, in its own right, a fascinating and

difficult problem. At the time of writing, many different platforms for quantum computing are

being actively developed or researched [19, 17, 65, 47], and after enormous investments [53] the

technology is improving at a steady clip. However, robust error-corrected quantum computing

remains out of reach. Nevertheless, this dissertation will, for sake of analysis, typically assume the

kind of noiseless (or error corrected) quantum computing capabilities that we hope for in the future.

Occasional comments on the likely effects of hardware noise are made, but detailed analysis is left

for future work.

2.5 Quantum Hamiltonians: A Closer Look

Having motivated digital quantum Hamiltonian simulation, let’s now elaborate on what this

entails. First, strictly speaking, Hamiltonians describe closed quantum systems. Closed systems

are idealizations of any real physical system, where the system of interest is perfectly isolated from

any external influence. Though this is a useful concept in all areas of physics, closed systems

are in some sense far from typical. Indeed, the science of decoherence suggests that the apparent

classicality of our world emerges from open system dynamics [143]. For open systems, a more

general framework of quantum states and operations needs to be invoked, but for our purposes the

mathematical representations we’ve discussed thus far (state vectors, unitary operations, etc.) will

suffice.

Taking physical time 𝑡 to be continuous, we imagine the state |𝜓𝑡⟩ at any time 𝑡 to be related to

the state at a previous time 𝑠 ≤ 𝑡 via some unknown unitary operation, which we denote 𝑈 (𝑡, 𝑠).

|𝜓𝑡⟩ = 𝑈 (𝑡, 𝑠) |𝜓𝑠⟩

(2.11)

What properties should 𝑈 possess? Tacit in our notation is the assumption that 𝑈 does not depend on

the states |𝜓𝑠⟩ and |𝜓𝑡⟩, meaning that the dynamical laws themselves do not care about the specific

states involved. This is also the situation we find in classical mechanics. An another reasonable

property, we might expect various 𝑈 to chain together naturally when applied in succession.

4Part of the solution likely entail better protocols for correcting errors.

𝑈 (𝑡, 𝑠) = 𝑈 (𝑡, 𝑟)𝑈 (𝑟, 𝑠)

𝑠 ≤ 𝑟 ≤ 𝑡

(2.12)

16

As a reasonable corollary, we have 𝑈 (𝑡, 𝑡) = 𝐼. It is sensible to define 𝑈 (𝑠, 𝑡) ≡ 𝑈 (𝑡, 𝑠)† for 𝑠 < 𝑡,

and with this the transitive property above generalizes to any 𝑠, 𝑟, 𝑡 ∈ R.

We might reasonably impose some degree of continuity to the wavefunction |𝜓𝑡⟩, hence to the

unitary 𝑈. Let’s assume, at the very least, that 𝑈 (𝑡, 𝑠) is differentiable in 𝑡 (and by symmetry, 𝑠).

Taking a derivative of equation (2.11) with respect to 𝑡, we obtain

(cid:12)
(cid:12)𝜓′
𝑡

(cid:11) = 𝑈′(𝑡, 𝑠) |𝜓𝑠⟩

= 𝑈′(𝑡, 𝑠)𝑈 (𝑠, 𝑡) |𝜓𝑡⟩ .

(2.13)

Now we have a differential equation in |𝜓𝑡⟩. For consistency, we must have that 𝑈′(𝑡, 𝑠)𝑈 (𝑠, 𝑡) is

independent of 𝑠 (this can be verified explicitly). Moreover, by using some basic properties of 𝑈

and the product rule, it is easy to check that it is also anti-Hermitian. Thus, if

−𝑖𝐻 (𝑡) ≡ 𝑈′(𝑡, 𝑠)𝑈 (𝑠, 𝑡),

(2.14)

then 𝐻 is a Hermitian operator. We call 𝐻 the Hamiltonian, and with this notation we recover the

famous Schrödinger equation

𝑖𝜕𝑡 |𝜓𝑡⟩ = 𝐻 (𝑡) |𝜓𝑡⟩ .

More generally, using relation (2.11), we obtain an operator Schrödinger equation

𝑖𝜕𝑡𝑈 (𝑡, 𝑠) = 𝐻 (𝑡)𝑈 (𝑡, 𝑠)

(2.15)

(2.16)

which, in conjunction with (2.11), provides an expression for a quantum state at any time 𝑡 following

an initial time 𝑠.

To the practitioner of quantum physics, this might seem like an odd approach to the foundations

of quantum dynamics. We started by postulating a unitary evolution operator 𝑈 with certain

reasonable properties, then derived a Hamiltonian which generates 𝑈 via the Schrödinger equation.

In physics, one typically starts with a Hamiltonian, then attempts to solve for 𝑈. It is insightful

to ponder the reasons for this. As discussed in Section 2.1, the Hamiltonian concept is closely

linked to classical mechanics, and hence serves as a more natural starting point for humans to

model quantum phenomena. Perhaps more fundamentally, a compact, elementary description of

17

𝑈 instead of 𝐻 would be like having the cheat code for solving any quantum physics problem. By

analogy, while the forces on classical systems may be more or less easy to describe, predicting

resulting trajectories is a straightforward but expensive computational task.

In short, having 𝑈

instead of 𝐻 feels "too good to be true."

The primary problem of quantum simulation, then, is to solve (2.16), or (2.15) more specifically,

given a description of 𝐻 (𝑡). As needed for sensible physics, the solution for 𝑈 (𝑡, 𝑠) exists and is

unique with the initial condition 𝑈 (𝑠, 𝑠) = 𝐼. More interesting is that a succinct description of the

solution exists. We may write it as a so-called time ordered operator exponential

𝑈 (𝑡, 𝑠) = expT

∫ 𝑡

(cid:26)

−𝑖

𝑠

(cid:27)

𝐻 (𝜏)𝑑𝜏

(2.17)

which may be understood in a number of ways. One that is particularly relevant to this thesis is

the product integration approach. Given a family of partitions {𝑡 𝑗 }𝑛

𝑗=1 of the interval [𝑠, 𝑡], with

maximum width 𝛿𝑛 tending to zero as 𝑛 → ∞, a solution is given by the product integral [73]

𝑈 (𝑡, 𝑠) = lim
𝑛→∞

𝑛−1
(cid:214)

𝑗=1

𝑒−𝑖𝐻 (𝑡 𝑗 )𝛿𝑡 𝑗

(2.18)

where 𝛿𝑡 𝑗 = 𝑡 𝑗+1 − 𝑡 𝑗 . One feature of this approach is that, for sufficiently large but finite 𝑛, the

product represents an approximation that is also unitary, in contrast with the more common Dyson

series representation. Solution (2.18) is closely linked to the idea of product formulas, which will

be discussed in subsequent chapters. An even simpler expression for 𝑈 can be found when 𝐻 is

independent of time. Philosophically, this condition amounts to the notion that the laws of physics

should not change over time. In this case, (2.18) simplifies to a simple operator exponential

𝑈 (𝑡, 𝑠) = 𝑒−𝑖𝐻Δ𝑡

(2.19)

where Δ𝑡 = 𝑡 − 𝑠. This expression can understood through a power series expansion or the spectral

theorem for normal operators. It is remarkable that such a simple and succinct expression can be

written for the solution to essentially all closed quantum dynamics. If 𝐻 is over a finite-dimensional

space, i.e., a matrix, computing a partial sum

𝑈 (𝑡, 𝑠) ≈

(−𝑖𝐻Δ𝑡) 𝑗
𝑗!

𝑁
∑︁

𝑗=0

18

(2.20)

for a sufficiently large 𝑁 will yield an arbitrarily accurate approximation of 𝑈 (𝑡, 𝑠). Hence,

given the matrix 𝐻, the computation of 𝑈 reduces to "just" matrix multiplication and addition.

Similarly, equation (2.18) can be approximated by taking 𝑛 sufficiently large (but finite) and

calculating the product of matrix exponentials. For systems that aren’t finite dimensional, they

may nevertheless be approximated to arbitrary accuracy by a sufficiently large, but finite, quantum

system via discretization. Consistent with our previous discussion, we have shown that quantum

dynamics may be computed using standard computational techniques, however, the exponentially

large matrices involved ensure that the above approach will not be efficient.

In light of the elegant solutions to the Schrödinger equation given by expressions such (2.19),

and more generally (2.18), a mathematically inclined reader may conclude that closed-system

quantum dynamics is, at the broadest level, a solved problem. However, without computational

methods, we cannot extract useful information from these solutions, such as what’s needed to make

concrete predictions about the behavior of a physical system. In this section, our goal is to clearly

state a high-level procedure for carrying out such computations on both classical and quantum

computers.

2.6 Classical Simulation Algorithms

We’ve already discussed the obstacles to classical simulation of quantum mechanics in prior

sections. Despite these, the importance of the problem to physics, and more broadly natural science,

has led practitioners to develop very clever methods that provide insight in limited but interesting

cases. Exact diagonalization [86] of the Hamiltonian, expressed as a matrix in a suitable basis,

is a guaranteed approach in principle but intractable for large systems. Quantum Monte Carlo [8]

methods refer to a broad range of techniques which utilize random sampling, and are especially

effective at calculating the low lying energies of a Hamiltonian. However, these cannot be used

directly for general calculations of quantum dynamics. Moreover, Monte Carlo techniques suffer

a notorious sign problem, in which large quantities of various signs need to be added together to

get a relatively small result. This leads to severe round-off errors from floating-point arithmetic.

Indeed, it would be both interesting and surprising if Monte Carlo methods were more successful

19

at computing quantum properties. This would suggest that, computationally, the randomness of

quantum theory could be reduced to mere coin flips. For this author, the sign problem is an

indication that there is more to quantum probability theory than normal randomness.

More recently, tensor network methods, based on Matrix Product States (MPSs) or Projected

Entangled Pairs (PEPs), represent the state of the art for dynamical simulation. As the name

suggests, such methods involve the representation of quantum states as tensor networks rather than

a vector of amplitudes. For systems with low entanglement, the "rank" of the tensors involved is not

too large. This allows for efficient representation and manipulation of quantum states. Area-law

bounds on entanglement, as found in lattice systems with geometrically local interactions, aid the

effectiveness of such methods [43]. Moreover, noise in imperfectly isolated computers can reduce

the coherence and entanglement, which can be further exploited. Thus, tensor networks have

brought more quantum systems within the capabilities of classical computers, and have become a

standard benchmark for testing the classical feasibility of quantum circuits.

Despite the aforementioned techniques, efficient simulation by classical means is likely unob-

tainable. This comes from both complexity theoretic arguments as well as the sheer effort towards

making effective simulation methods.

2.7 Quantum Simulation Algorithms

As we’ve anticipated, quantum computers provide a platform for efficient quantum simulation

given reasonable assumptions on the input Hamiltonian. Figure 2.2, reprinted from [52], gives

a high-level overview of the Hamiltonian simulation workflow on quantum devices. Before any

computation can be performed, the problem of interest must be encoded onto a quantum computer.

This could be accomplished in several ways depending on the nature of the problem. For example,

effective mappings from fermionic [94] and bosonic [116] systems to qubits have been extensively

studied in the literature. We will not say much more on how to properly map a problem of

interest onto a collection of qubits, though this is a crucial starting step for any attempt at quantum

Hamiltonian simulation.

Having identified a mapping, one can turn to the simulation proper. An initial state |𝜓0⟩

20

Figure 2.2 Schematic of the quantum simulation workflow. The system of interest, represented by
the upper "cloud," is first represented on the quantum computer through some correspondence
given by the dashed lines. The three main tasks for simulation are state preparation |𝜓(0)⟩, time
evolution by 𝑈 (𝑡, 0), and a measurement procedure. Each of these steps has its own set of tools
and methods. Reprinted from [52] with permission. Copyright 2014 by the American Physical
Society.

of the simulation must be prepared by a quantum circuit given an initial "fiducial state" of the
device, typically |0⟩⊗𝑛

. The difficulty of preparing |𝜓0⟩ depends on the nature of the problem

and the qubit encoding used. Entire subdomains of digital Hamiltonian simulation are devoted

to preparing effective initial states in various contexts [121, 40, 9, 99]. Although special states

such as ground states and wavepackets are often sought for, the arbitrariness of initial state choice

makes it difficult to do a general analysis of such methods. In contrast, given a specific input model

for the Hamiltonian 𝐻 (𝑡), one can express general algorithms for generating a quantum circuit

𝑉 which approximates the time evolution operator 𝑈. Often, this step alone is what is referred

to as "Hamiltonian simulation", though it is necessarily one part of the full process needed to

extract useful information. Essentially, the problem is, given an input Hamiltonian 𝐻 (𝑡), a desired

simulation interval [0, 𝑡] and a desired tolerance 𝜖 > 0 construct a quantum circuit 𝑉 such that the

output 𝑉 |𝜓0⟩ is within 𝜖 of 𝑈 (𝑡, 0) |𝜓0⟩.

Once the state is evolved in time, it needs to be measured. Simply having the evolved state on a

quantum register is not enough, and is nothing like having the state vector written on paper. Learning

21

the full output state |𝜓𝑡⟩, known as full-state tomography, is extremely inefficient. However, in

actual quantum calculations we are typically interested in a handful {𝑂𝑖} of observables of interest.

Learning these observables given |𝜓𝑡⟩ turns out to be a much more approachable task; standard

routines such as phase estimation and amplitude estimation are sufficient [56, 67]. In Section 6.3,

we will discuss the Rodeo Algorithm, a new addition to the suite of phase estimation algorithms

which can perform resource-efficient measurements in the eigenbasis of a Hamiltonian using time

evolution.

Despite the utility of the of the Preparation-Evolution-Measurement schematic, actual simula-

tion protocols may not be purely sequential. For example, optimal protocols for eigenvalue and

expectation value measurement protocols, particularly quantum phase estimation protocols, incor-

porate the Evolution as a subroutine. We will in fact see this in the Iterative Amplitude Estimation

protocol of Chapter 3. Even so, the framework is still useful in providing a program for developing

simulation algorithms. In any conceivable case where a simulation is needed, we will need a way

to (a) prepare a state, (b) evolve the state and (c) measure the state.

Without additional assumptions, even quantum computers cannot solve Hamiltonian simulation

efficiently in all instances. A simple counting argument shows that an 𝜖-approximation to an 𝑛

qubit unitary 𝑈 requires, in general, an exponentially large number of elementary quantum gates

in 𝑛 [101]. Any such unitary may, if desired, be viewed as a time evolution operator for some

Hamiltonian 𝐻.

𝑈 = 𝑒−𝑖𝐻

(2.21)

Evidently, there are some Hamiltonians 𝐻 whose simulation is requires exponential quantum

resources, and is hence intractable.

This point aside, most Hamiltonians of interest are not of this nature. Physical Hamiltonians, as

described previously, are naturally 𝑘-local, and we shall leverage this in the product formula algo-

rithms below. In contexts removed from physics, such as solving linear systems, the Hamiltonian

𝐻 is a sparse matrix (with efficiently computable and locatable nonzero entries). Such assumptions

ensure efficient simulations are possible, and are typically baked into the definition of "Hamiltonian

22

simulation" and not mentioned. Over the past several decades, enormous progress has been made

such that there are multiple good quantum algorithms for approximating 𝑈 (𝑡, 𝑠). We now outline

the three categories which, broadly speaking, classify all such methods.

2.7.1 Product Formulas

Early thinkers like Feynman and Deutsch had long claimed that quantum computers could

efficiently simulate quantum mechanics, but it was Seth Lloyd’s seminal algorithm, based on

Trotterization, that first proved this was the case [87]. Lloyd considered 𝑘-local Hamiltonians of

the form (2.9), and in this case the exponentials exp{−𝑖𝐻𝛾Δ𝑡} of each term were unitary operations

on only 𝑘 qubits. Since 𝑘 remains fixed as the number of qubits 𝑛 grew, these exponentials can

be implemented as quantum circuits of fixed depth. Moreover, these unitaries can be combined in

sequence to approximate the full time evolution exp{−𝑖𝐻𝑡}. In his paper, Lloyd used the so-called

first-order Trotter formula 𝑆1, defined as

𝑆1(𝑡) := 𝑒−𝑖𝐻1𝑡𝑒−𝑖𝐻2𝑡 . . . 𝑒−𝑖𝐻Γ𝑡 .

This is a unitary operator which for small 𝑡, accurately approximates 𝑈 (𝑡). In particular,

𝑆1(𝑡) − 𝑒−𝑖𝐻𝑡 ∈ 𝑂 (𝑡2),

(2.22)

(2.23)

where 𝑡 is taken asymptotically to zero. Longer simulation times can be achieved by dividing the

full interval into 𝑟 steps.

𝑆1(𝑡/𝑟)𝑟 =

(cid:16)

𝑒−𝑖𝐻1𝑡/𝑟 𝑒−𝑖𝐻2𝑡/𝑟 . . . 𝑒−𝑖𝐻Γ𝑡/𝑟 (cid:17)𝑟

= 𝑒−𝑖𝐻𝑡 + 𝑂 (𝑡2/𝑟)

(2.24)

By taking 𝑟 sufficiently large, the error can be arbitrarily diminished.

More generally, product formulas are unitary approximations to 𝑒−𝑖𝐻𝑡 made by splitting the

exponential along the terms 𝐻𝛾 in a specified sequence. The order of a product formula P

characterizes the degree of approximation to 𝑈, and is the largest integer 𝑝 ∈ Z+ such that

𝑈 (𝑡) − P (𝑡) ∈ 𝑂 (𝑡 𝑝+1).

(2.25)

In other words, the order is the largest 𝑝 for which 𝑈 (𝑡) and P (𝑡) share the same 𝑝th Taylor

polynomial. This definition justifies our referring to (2.22) as 1st order. Product formulae exist for

23

all orders 𝑝, and in fact there is a recursive procedure for generating higher order formulas from

lower ones [123]. The standard candle is the so-called Suzuki-Trotter formulas. To define these let

𝑆1 be given by (2.22), and define

𝑆2(𝑡) := rev[𝑆1(𝑡/2)]𝑆1(𝑡/2)

(2.26)

where rev is the reverse of the terms in the product of 𝑆1. It can be quickly verified via Taylor

expansion that 𝑆2 is second order. It is also symmetric, both in the sense of the ordering of its terms

and in a time reversal sense.

𝑆2(−𝑡) = 𝑆2(𝑡)†

(2.27)

Any product formula P satisfying the condition (2.27) will be termed "symmetric." Symmetric

product formulas have the useful property that their error series, namely the power series of

𝑈 (𝑡) − P (𝑡), is an odd function. Thus, any procedure which seeks to eliminate errors term by term

can skip all even powers.

For any 𝑘 ∈ Z+, with 𝑘 > 1, we define 𝑆2𝑘 recursively as

𝑆2𝑘 (𝑡) = 𝑆2

2(𝑘−1) (𝑢𝑘𝑡)𝑆2(𝑘−1) ((1 − 4𝑢𝑘 )𝑡)𝑆2

2(𝑘−1) (𝑢𝑘𝑡),

(2.28)

where 𝑢𝑘 = (4 − 4(𝑘−1))−1. This formula is symmetric and order 2𝑘. Thus, product formulas of

arbitrary order exist. However, we observe that our recursive procedure generates an exponentially

increasing number of unitaries as a function of 𝑘, leading to impractically high costs for modest

accuracy gains. It is possible to show that this feature is present for any high order formula, by

considering the number of terms needed to eliminate errors term by term. Thus, in practice, only

the lowest orders formulas are used. Nevertheless, the existence of arbitrary order formulas is

valuable theoretically to understand asymptotic scaling of simulation costs. Moreover, the forward

and backward evolutions present in (2.28) give rise to fractal behavior for large 𝑘 that is itself

interesting [122].

Product formulas satisfy the useful property that, in the case where all the 𝐻𝛾 commute, the

error in the Trotterization vanishes, for the same reason that 𝑒𝑤𝑒𝑧 = 𝑒𝑤+𝑧 for 𝑤, 𝑧 ∈ C. More

24

generally, one should expect the simulation errors to be small even when the commutator

[𝐻𝑖, 𝐻 𝑗 ] := 𝐻𝑖𝐻 𝑗 − 𝐻 𝑗 𝐻𝑖

(2.29)

is small but nonzero. We term this feature commutator scaling. It took a surprisingly long time

to show this rigorously, but was finally done using the calculus of matrix-valued functions [27].

These results applied to a general class of staged product formulas of the form

P (𝑡) =

Υ
(cid:214)

Γ
(cid:214)

𝑗=1

𝛾=1

𝑒−𝑡𝜏𝑗 𝛾 𝐻 𝜋 𝑗 (𝛾) .

(2.30)

Here, Υ is the number of "stages", and 𝜋 𝑗 is a permutation of the first Γ positive integers. Moreover,

𝜏𝑗 𝑘 are real numbers. The Suzuki-Trotter formulas are examples of staged formulas. It was shown

that the additive and multiplicative errors A, M in the estimation of 𝑒−𝑖𝐻𝑡 for a 𝑝th order staged

product formula P𝑝 scale as

A, M ∈ 𝑂 ( ˜𝛼comm𝑡 𝑝+1)

where

˜𝛼comm =

Γ
∑︁

Γ
∑︁

· · ·

Γ
∑︁

𝛾1=1

𝛾2=1

𝛾 𝑝+1=1

∥ [𝐻𝛾 𝑝+1

, . . . , [𝐻𝛾2

, 𝐻𝛾1], . . . ] ∥

(2.31)

(2.32)

is a sum of the norms of all nested commutators of 𝑝 + 1 terms of 𝐻 [27]. Though ˜𝛼comm may be

practically difficult to compute, this result gives a tighter characterization of the kinds of errors to

expect from a Trotter simulation. Moreover, these commutators can be computed once for specific

classes of Hamiltonians, such as lattice systems, and the results can be applied thereafter.

Despite these sophisticated theoretical characterizations, product formulas have been observed

to perform even better than expected [25, 66]. The relative simplicity and flexibility of Trotter meth-

ods, as compared to the more recent and theoretically improved methods described in subsequent

sections, makes this approach the current frontrunner in practical quantum simulation. However, as

quantum hardware continues to improve, we may find that so-called post-Trotter methods become

increasingly attractive as the overhead costs become less burdensome.

25

Using the error bounds (2.31), we can derive a rigorous upper bound on the number of Trotter

steps 𝑟 needed to simulate a 𝑘-local 𝐻 for time 𝑇 to accuracy 𝜖 using a 𝑝th order product formula.

𝑟 ∈ 𝑂

(cid:32)

˜𝛼1/𝑝
comm𝑇 1+1/𝑝
𝜖 1/𝑝

(cid:33)

(2.33)

For fixed order formula 𝑝 the total number of exponentials 𝑒−𝑖𝐻 𝑗 𝛿𝑡 scales as 𝑟 up to constant factors.

Since each exponential requires at most a constant number of quantum gates, the above formula

also gives the scaling of the number of two-qubit gates. What we see is that we have achieved

efficient simulation in terms of 𝑇 and 𝜖. Moreover, the number of qubits needed is only those 𝑛

which generate the state space of the system. Contrast this with the 2𝑛 needed, naively, to write the

full state classically.

Product formulas will recur throughout this thesis. One of the deficiencies of this approach

to simulation is their relatively low accuracy, especially to post-Trotter methods with 𝑂 (log 1/𝜖)

scaling in the accuracy. In Chapter 3, we explore the use of polynomial interpolation to improve this

accuracy without employing additional quantum resources. Trotterization also arises in the time

dependent setting, where each 𝐻𝛾 (𝑡) depends on time, by using (2.18) for finite 𝑛 and Trotterizing

each 𝐻 (𝑡 𝑗 ). The use of an auxiliary "clock space" connects the notions time independent and time

dependent simulation (and Trotterization), which we use to propose and analyze several approaches

to time dependent Hamiltonian simulation. This is considered in Chapter 4.

2.7.2 Linear Combination of Unitaries

For a couple decades (a long time relative to the field), product formulas were the only algorithm

in town for Hamiltonian simulation on quantum computers. Then, in 2012, Childs and Wiebe

introduced [24] a new primitive for quantum computation: applying a linear combination of

unitaries

𝐿
∑︁

𝑗=1

𝛼 𝑗𝑈 𝑗 .

(2.34)

to a quantum register. Figure 2.3 gives a schematic circuit for implementing this sum. Because

a sum of unitaries is not unitary, we should expect that some measurements are required for the

successful implementation of the operation, and indeed, it is conditioned on measuring all 0’s on

26

|0⟩

|0⟩

|0⟩

|𝜓⟩

PREP†

. . .

. . .

. . .

. . .

𝑈8

PREP

𝑚

𝑈1

𝑈2

Figure 2.3 A schematic of the linear combination of unitaries (LCU) circuit. Conditioned upon
measuring 0 on every measurement, the result applied is (cid:205) 𝑗 𝛼 𝑗𝑈 𝑗 . In this case, there are 3
auxiliary qubits, hence 23 = 8 possible unitaries to add. The PREP circuits are any circuit
satisfying PREP |0⟩𝑛 = (cid:205) 𝑗

√𝛼 𝑗 | 𝑗⟩.

the auxiliary register. Measuring "success" is more likely the closer (2.34) is to a unitary operator.

Success can be achieved through repeated trials or, more efficiently, through quantum amplitude

amplification.

The linear combination of unitaries (LCU) circuit consists of two subroutines. The first is a

PREP ("prepare") unitary which acts on a quantum register initialized to |0⟩⊗𝑘

as

PREP |0⟩⊗𝑘 =

𝐿
∑︁

√︂ 𝛼 𝑗
∥𝑎∥1

| 𝑗⟩

(2.35)

𝑗=1
with ∥𝑎∥1 := (cid:205) 𝑗 |𝑎 𝑗 |. The second is a SEL ("select") unitary which applies 𝑈 𝑗 to the main register
controlled on state 𝑗 of the auxiliary.

SEL | 𝑗⟩ |𝜓⟩ = | 𝑗⟩ 𝑈 𝑗 |𝜓⟩

(2.36)

By applying the operation (PREP† ⊗ 𝐼)SEL(PREP ⊗ 𝐼), then measuring the appropriate outcome

on the auxiliary (all zeros), the operation (2.34) may be implemented on the main register up to

normalization.

The LCU primitive led to several distinct Hamiltonian simulation algorithms, the first post-

Trotter methods. Here we discuss the one most relative to this thesis, and the one considered in

the original Childs and Wiebe paper. This is the quantum implementation of the multiproduct

formula (MPF), which is a linear combination of product formulas producing a more accurate

approximation to 𝑒−𝑖𝐻𝑡. The best way to understand multiproduct formulas is as an instance of

27

Richardson extrapolation, and we will elaborate on this point in Chapter 5. The major point is that

we wish to extrapolate 𝑟 → ∞, or equivalently 1/𝑟 → 0, to achieve increasing accuracies. While

higher-order Trotter formulas require an exponential number of terms, it is much less demanding

to cancel error terms using summation. The upshot is that MPFs, and other well-known LCU

algorithms, achieve an exponential improvement in accuracy over product formulas alone.

2.7.3 Qubitization

In our discussion of the Linear Combination of Unitaries (LCU) primitive above, we saw that

the unitary

𝑈 = (PREP† ⊗ 𝐼)SEL(PREP ⊗ 𝐼)

(2.37)

encodes the desired operation (a sum of unitaries) provide a certain measurement result is achieved

on a part of the full quantum register. This turns out to point towards a more general phenomenon.
We say that 𝑈 block encodes the desired operation (cid:205) 𝑗 𝛼 𝑗𝑈 𝑗 via the state |0⟩⊗𝑘

, in the sense that

𝛼 𝑗𝑈 𝑗 = (⟨0|⊗𝑘 ⊗ 𝐼)𝑈 (|0⟩⊗𝑘 ⊗ 𝐼)

(2.38)

∑︁

𝑗

up to normalization. In our context, this block-encoded operator may be the Hamiltonian of interest.

Remarkably, this simple requirement, the encoding of 𝐻 in a subblock of a larger unitary matrix, is

all that is necessary for qubitization [88], the first asymptotically optimal approach to Hamiltonian

simulation. By this, we mean that the simulation cost for simulating for time 𝑇 and accuracy 𝜖

scales as

(cid:18)
𝑇 +

𝑂

(cid:19)

log 1/𝜖
log log 1/𝜖

(2.39)

which saturates the known lower bounds [14] for 𝑇 and 1/𝜖 and is additive rather than multiplicative.

Besides the concept of block encoding, the other major ingredient to qubitization is Quan-

tum Signal Processing (QSP), which generates polynomial functions of the eigenvalues of the

block encoded operator. It does so by interleaving unitary rotations that act independently on a

two-dimensional subspace corresponding to each eigenvalue. Since 𝑒−𝑖𝜆𝑡 can be approximated

by polynomials 𝑃(𝜆𝑡) uniformly over an interval [0, 𝜆𝑇], constructing these polynomial transfor-

mations provide a means to Hamiltonian simulation. Qubitization was later generalized to the

28

Quantum Singular Value Transformation (QSVT) [54], which performs polynomial transformation

of the singular values of a block encoded matrix. The method is effective, general, conceptually

rich and, unfortunately, rather complicated to understand fully.

One notable drawback to qubitization is that it only directly works for time independent Hamil-

tonian simulations. Of course, one could Trotterize via (2.18) and perform qubitization on each

factor, but the error from the Trotterization would overshadow any gains from qubitization.

In

Chapter 4, we will embed 𝐻 (𝑡) in a system with time dependent Hamiltonian, then apply qubitiza-

tion "directly." We construct a suitable block encoding of the augmented Hamiltonian, then derive

bounds on the query complexity. While our analysis does not show improvements compared to

other time dependent schemes, we believe this is from imperfections in the analysis rather than a

features of the method.

2.7.4 Analog Quantum Simulation

In order to anticipate the material of Section 6.1, we digress momentarily from our discussion of

quantum computers and consider an alternative simulation platform: analog quantum simulators.

Recall from Figure 2.2 the need to map the problem of interest onto a simulator.

Instead of

mapping to a "digital" setting, in which a universal set of discrete, unitary quantum gates are

applied in sequence to construct the time evolution operator, we could instead map to a system

which, though not a quantum computer, nevertheless allows for a great deal of control and emulation

of a system of interest. This is potentially less technologically demanding, because we no longer

need to perform arbitrary unitary operations, but rather specific Hamiltonians of interest.

For excellent discussions of analog quantum simulators, see e.g. [63, 52]. For our purposes,

we can roughly model analog simulators as devices which can implement a class of Hamiltonians,

with terms

∑︁

𝐻 =

𝐽 𝑗 (𝑡)𝐻 𝑗

𝑗

(2.40)

such that the time dependence 𝐽 𝑗 (𝑡) is controllable to some degree. Depending on the collection

of 𝐻 𝑗 and the degree of time dependence 𝐽 𝑗 , we might be able to achieve a universal device. Yet

this may not be necessary for the application we are interested in.

29

Analog simulators are often considered a more achievable route to quantum Hamiltonian

simulation. In Section 6.1 we will explore some interesting phenomena accessible to trapped-ion

quantum simulators, with implications in nuclear physics.

2.7.5 Measurement and Hamiltonian Evolution

As already emphasized, time evolution represents only one piece of a full quantum simulation

algorithm. Although the time evolved state |𝜓𝑡⟩ is prepared on a quantum register, no information

is gained without measurement. By analogy, a random variable that is not sampled leaves nothing

gained. There is nontrivial work to be done in the extraction of information, as sampling in

the quantum case (i.e., measurement) is more nuanced than simple probability sampling. This

is because multiple different bases can be measured, choosing the right basis is important for

extracting valuable information.

One basic fact we might like to know about a Hamiltonian are its eigenvalues, which correspond

to the "allowed" energies of a physical system. Additionally, general observables 𝑂 can be often

simulated as if they were Hamiltonians assuming they are, say, sparse or 𝑘-local. Suppose we

are able to prepare an (approximate) eigenvector |𝐸⟩ of 𝐻 on a quantum register. Performing

Hamiltonian simulation will produce an output state that picks up merely a phase.

𝑈 (𝑡) |𝐸⟩ = 𝑒−𝑖𝐸𝑡 |𝐸⟩

(2.41)

This phase cannot be measured as is, because only relative phase shifts, not overall phases, are

measurable in general wavelike phenomena. Our strategy will be to introduce a reference unshifted

state and produce interference, such that the phase 𝜑 = 𝐸𝑡 is measurable. One way to do this in the

setting of quantum computing is to apply a controlled-𝑈 gate. By putting an auxiliary register in a

superposition |0⟩ + |1⟩ and applying 𝑈 to the main register conditioned on |1⟩, we’ve introduced a

relative phase shift into the auxiliary qubit. This can be measured through a change of basis.

What we’ve heuristically described above is a general framework for performing phase esti-

mation on a quantum computer, which allows for the extraction of eigenvalues. Figure 2.4 gives

the basic circuit for the simplest phase estimation algorithm. Physically, the circuit is essentially a

30

|0⟩

|𝐸⟩

Had

𝑛

𝑒−𝑖𝐻𝑡

𝑃(𝜃)

Had

Figure 2.4 Schematic of the simplest quantum phase estimation algorithm, which is useful for
measuring in the eigenbasis of a Hamiltonian. The measurement outcomes associated with the top
auxiliary qubit are directly related to the eigenvalues of 𝐻, and by repeated trials these eigenvalues
can be estimated. The phase rotation 𝑃(𝜃) has an angle parameter 𝜃 which may be varied to
resolve certain ambiguities due to cos2 not being one-to-one on [0, 𝜋].

Mach-Zehnder interferometer [67], with Hadamards

Had :=

1
√

2

1

1

(cid:169)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
1 −1
(cid:172)

(2.42)

acting as beam splitters and the controlled time evolution introducing a phase shift. The family

of phase estimation algorithms is, by now, enormous [67, 77, 125, 136]. This thesis will discuss

a recent addition to this family known as the Rodeo Algorithm, a resource-efficient, randomized

procedure which performs a selective search over the space of possible eigenvalues. Like all

phase estimation protocols, it works well provided the initial state has reasonable overlap with the

eigenstates of interest (formally, if the overlap decreases polynomially with problem size).

Perhaps even more foundational than phase estimation, in a sense, is the estimation of an

amplitude on a quantum computer.

In fact, the simple scheme above encodes the phase in an

amplitude to be measured. Slightly more generally, we suppose an operator 𝑉 acting on an initial

state |0⟩ of a single qubit as 𝑉 |0⟩ = 𝑎0 |0⟩ + 𝑎1 |1⟩, and wish to estimate 𝑎0 (without loss of

generality we may take 𝑎0 ≥ 0). This estimation may be done through repeated computational

basis measurements, but more clever approaches using amplitude amplification lead to a quadratic

improvement in scaling. See [59] for details on an efficient iterative procedure for amplitude

estimation.

2.8 Mathematical Reference

The following section includes important technical definitions and results that are used through-

out the dissertation. The reader may, if interested, browse these mathematical tools now, or they

may come back and reference as needed as they read subsequent chapters.

31

2.8.1 Combinatorics

The simple factorial 𝑛! counts the number of permutations of 𝑛 objects, and is usefully ap-

proximated by Stirling’s approximation.

In the paper, we always make use of a version of the

approximation which gives strict bounds for 𝑛 ∈ Z+.

√

2𝜋𝑛

(cid:17) 𝑛

(cid:16) 𝑛
𝑒

< 𝑛! <

√

2𝜋𝑛

(cid:17) 𝑛

(cid:16) 𝑛
𝑒

𝑒1/(12𝑛)

(2.43)

These bounds are extremely tight, even for small 𝑛.

The multinomial coefficient is a generalization of the more common binomial coefficient, and

it arises in several combinatorial situations. It is defined by

(cid:19)

(cid:18)

𝑛
𝑛1, ..., 𝑛𝑘

:=

𝑛!
𝑛1!𝑛2!...𝑛𝑘 !

(2.44)

where 𝑛 ∈ Z+ and the (𝑛ℓ) 𝑘

ℓ=1 are nonnegative integers which sum to 𝑛. It is a positive integer
corresponding to the number of distinct ways of placing 𝑛 distinguishable items into 𝑘 boxes, where

each box has a fixed number 𝑛ℓ of items. In this work, we will find occasion to make use of the

multinomial when evaluating high-order derivatives of a product.

(cid:19) 𝑛

(cid:18) 𝑑
𝑑𝑡

𝑓1(𝑡) 𝑓2(𝑡) . . . 𝑓𝑘 (𝑡)

(2.45)

Here, ( 𝑓ℓ) 𝑘

ℓ=1 are 𝑛-differentiable functions of 𝑡 ∈ R. Employing the product rule, one is left to
count all the possible combinations of derivatives of each 𝑓ℓ. It turns out that the multinomial is

suited for this.

(cid:18) 𝑑
𝑑𝑡

(cid:19) 𝑛 𝑘
(cid:214)

ℓ=1

𝑓ℓ (𝑡) =

(cid:18)

∑︁

𝑁

𝑛
𝑛1, . . . , 𝑛𝑘

(cid:19)

𝑘
(cid:214)

ℓ=1

(cid:19) 𝑛ℓ

(cid:18) 𝑑
𝑑𝑡

𝑓ℓ (𝑡)

(2.46)

The sum is taken over the set 𝑁 of sequences of nonnegative integers (𝑛ℓ) 𝑘

ℓ=1 summing to 𝑛. A

useful property is that

(cid:18)

∑︁

𝑁

(cid:19)

𝑛
𝑛1, . . . , 𝑛𝑘

= 𝑘 𝑛

(2.47)

for nonnegative integers 𝑘, 𝑛 (with convention 00 = lim𝑥→0 𝑥𝑥 = 1).

Besides derivatives of products, we will also need to bound derivatives of ordinary exponentials

of a time dependent matrix. Useful for this purpose is an expression for derivatives of exponentials

32

of a scalar function 𝑎(𝑡).

(cid:19) 𝑛

(cid:18) 𝑑
𝑑𝑡

𝑒𝑎(𝑡)

The solution we rely on is Faà di Bruno’s formula, which asserts that

(cid:19) 𝑛

(cid:18) 𝑑
𝑑𝑡

𝑒𝑎(𝑡) = 𝑒𝑎(𝑡)𝑌𝑛 (𝑎′(𝑡), 𝑎”(𝑡), . . . , 𝑎 (𝑛) (𝑡))

where 𝑌𝑛 is the complete exponential Bell polynomial [31]. An explicit formula is given by

𝑌𝑛 (𝑥1, 𝑥2, . . . , 𝑥𝑛) =

∑︁

𝐶

𝑛!
𝑐1!𝑐2! . . . 𝑐𝑛!

(cid:19) 𝑐 𝑗

𝑛
(cid:214)

𝑗=1

(cid:18) 𝑥 𝑗
𝑗!

where the sum is taken over the set 𝐶 of all sequences (𝑐 𝑗 )𝑛

𝑗=1 such that 𝑐 𝑗 ≥ 0 and

𝑐1 + 2𝑐2 + · · · + 𝑛𝑐𝑛 = 𝑛.

(2.48)

(2.49)

(2.50)

(2.51)

Essentially, each coefficient in 𝑌𝑛 counts the ways one can partition a set of fixed size 𝑛 into subsets

of given sizes and number. When one simply wants to count the total number of possible partitions,

one is led to the Bell numbers 𝑏𝑛. These are related to the 𝑌𝑛 by evaluating all arguments to 1.

More generally, for any 𝑥 ∈ R,

𝑏𝑛 = 𝑌𝑛 (1, 1, ...1)

𝑌𝑛 (𝑥, 𝑥2, . . . , 𝑥𝑛) = 𝑥𝑛𝑏𝑛,

(2.52)

(2.53)

which can be seen directly from (2.50) along with the sum rule (2.51). The Bell numbers 𝑏𝑛 grow

combinatorially; in particular, the following upper bound [13] is useful.

𝑏𝑛 <

(cid:19) 𝑛

(cid:18)

.792𝑛
log(𝑛 + 1)

, ∀𝑛 ∈ Z+

(2.54)

More generally, the single-variable Bell polynomial, or Touchard polynomial 𝐵𝑛 (𝑥), is simply 𝑌𝑛

with all arguments evaluated to 𝑥.

𝐵𝑛 (𝑥) = 𝑌𝑛 (𝑥, 𝑥, . . . , 𝑥)

(2.55)

Of course, 𝑏𝑛 = 𝐵𝑛 (1). The 𝑛th Bell polynomial 𝐵𝑛 (𝑥) is also the value of the 𝑛th moment of the

Poisson distribution with mean 𝑥. From [5] we have the following upper bound on 𝐵𝑛
(cid:18)

(cid:19) 𝑛

𝐵𝑛 (𝑥) ≤

𝑛
log(1 + 𝑛
𝑥 )

, ∀𝑥 ≥ 0

(2.56)

33

which we observe is very close to that for the Bell numbers (𝑥 = 1) in equation (2.54). From

their definitions, 𝑌𝑛, 𝐵𝑛 and 𝑏𝑛 all grow monotonically, both in their functional arguments and their

index 𝑛. This is intuitive from being combinatorial functions whose coefficients count something

according to the size of 𝑛.

2.8.2 Norms

Norms are used widely throughout the paper to characterize the size of mathematical objects

and quantify simulation costs. For finite-dimensional vectors 𝑣 = (𝑣1, . . . , 𝑣𝑛), real valued or

complex, the Schatten 𝑝-norm, is defined as

1/𝑝

(2.57)

|𝑣| 𝑝 := (cid:169)
(cid:173)
(cid:171)

𝑛
∑︁

𝑗=1

|𝑣 𝑗 | 𝑝(cid:170)
(cid:174)
(cid:172)

for any 𝑝 ∈ [1, ∞), and for 𝑝 = ∞ as ∥𝑣∥∞ = max 𝑗 |𝑣 𝑗 |. We make particular use of the 1 and ∞

norm in our paper, to express our results or quote previous ones.

We also make use of functional norms, which are defined analogously. Given a scalar function

𝑓 (𝑡) with scalar input over an interval [0, 𝑇], the 𝑝-norm, or 𝐿 𝑝 norm, for 𝑝 ∈ [1, ∞) is given by

∥ 𝑓 (𝑡) ∥ 𝑝 :=

(cid:19) 1/𝑝

| 𝑓 (𝜏)| 𝑝𝑑𝜏

(cid:18)∫ 𝑇

0

(2.58)

for functions such that it is defined. Analogously, the ∞-norm is given by the supremum sup| 𝑓 (𝑡)|

over [0, 𝑇]. For the piecewise smooth functions we consider, this is just the maximum value on the

interval, so we might write ∥ 𝑓 (𝑡)∥max.

In Table 1, we use notation ∥ · ∥ 𝑝,𝑞 to denote nested norms whenever our objects have

both a (finite) vector and functional character. Specifically, this notation means take the vec-

tor 𝑝-norm first, then take the 𝑞-norm of the resulting scalar function. For example, if 𝛼(𝑡) =

(𝛼1(𝑡), 𝛼2(𝑡), . . . , 𝛼𝑛 (𝑡)), then

and

∥𝛼∥1,1 =

(cid:13)
𝑛
(cid:13)
∑︁
(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑗=1

|𝛼 𝑗 (𝑡)|

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)1

=

∫ 𝑇

𝑛
∑︁

0

𝑗=1

|𝛼 𝑗 (𝜏)|𝑑𝜏

∥𝛼∥1,max = max
𝜏∈[0,𝑇]

𝑛
∑︁

𝑗=1

|𝛼 𝑗 (𝜏)|.

34

(2.59)

(2.60)

In the main paper we claim our Hamiltonian simulation algorithm exhibits 𝐿1-norm scaling. This

means it has complexity 𝑂 (∥ 𝑓 ∥1), where 𝑓 is a function whose value is some measure of the size

of the Hamiltonian and its derivatives at each 𝑡 ∈ [0, 𝑇].

Finally, our paper makes use of the spectral norm for linear operators, also known as the induced

2-norm. It is defined for any bounded operator 𝐴 on a Hilbert space H by

∥ 𝐴∥ := sup

𝑣∈H \{0}

∥ 𝐴𝑣∥2
∥𝑣∥2

.

(2.61)

In our case, 𝐴 will always be finite dimensional, and ∥ 𝐴∥ is the largest singular value of 𝐴. This

norm is invariant under left or right multiplication by a unitary operator 𝑈, and ∥𝑈 ∥ = 1. The

spectral norm is submultiplicative, a property we make frequent use of.

∥ 𝐴𝐵∥ ≤ ∥ 𝐴∥∥𝐵∥

(2.62)

35

CHAPTER 3

TROTTER ERROR MITIGATION

This chapter is based on recent work [113] concerning the reduction of Trotter error by use of

standard Chebyshev interpolation, and is outlined as follows. After providing some background and

motivation, we spend two sections describing the interpolation procedure and proving some results

on its robustness to noisy data. We then apply the framework to the important task of measuring

expectation values of time-evolved observables. Numerical demonstrations are provided for a

random 1D Heisenberg model, and show the expected behavior. Discussion on the implications of

this work is given, and technical results are proven at the end of the chapter. With the exception of

the numerics of Section 5.7, which are new, additional details, applications, and numerics can be

found in the main publication [113].

3.1

Introduction and Motivation

In Section 2.7, we gave an overview of product formulas as a means for Hamiltonian simulation.

Efficient, versatile, and simple, product formulas will perhaps remain the preferred method of

quantum simulation on quantum computers for the foreseeable future. This motivates the search for

techniques to further bolster the method, particularly by mitigating its biggest flaws. Chief among

these, perhaps, is their relative inaccuracy compared to post-Trotter methods such as qubitization.
As an example, the 1st order Trotter formula 𝑆1, which splits the exponential 𝑒−𝑖𝐻𝑡 in the simplest

imaginable way,

𝑆1(𝑡) = 𝑒−𝑖𝐻1𝑡𝑒−𝑖𝐻2𝑡 . . . 𝑒−𝑖𝐻Γ𝑡

(3.1)

has an error scaling given by the 𝑂 (𝑡2/𝑟). From this we deduce that the simulation cost scales

with the error 𝜖 as 𝑂 (1/𝜖). Contrast this with, say, qubitization, which scales slightly better than

𝑂 (log 1/𝜖), an exponential improvement. More generally, every major post-Trotter method scales

polylogarithmically in 1/𝜖 (that is, 𝑂 ( 𝑝(log 1/𝜖)) for some polynomial 𝑝). Higher order product

formulas have better accuracy, but are generally impractical due to exponential scaling of cost with

the order 𝑘 of the formula.

Can the accuracy of product formulas be improved with additional techniques? Indeed, multi-

36

product formulas, mentioned in Section 2.7.2 and elaborated in Chapter 5, achieve exponentially

improved accuracy compared to product formulas alone by summing formulas of different Trot-

ter step size. Unfortunately, the additional quantum overhead and controlled operations required

for implementing the required LCU procedure are noteworthy, and introduce barriers which are

especially burdensome in the current era of noisy hardware.

Faced with these limitations, we might look beyond multiproduct formulas based on LCU

and ask if there is a way forward using only classical resources, such as randomness.

Indeed,

within the past few years there have been claims of using multiproduct formulas without LCU-type

procedures [45, 130]. These schemes don’t produce a true multiproduct formula, in the sense of

applying the operation

MPF(𝑡) =

𝐿
∑︁

𝑐 𝑗 P (𝑡/𝑟 𝑗 )𝑟 𝑗

(3.2)

𝑗=1
to a quantum register, where P is a product formula, 𝑟 𝑗 ∈ Z+, and 𝑐 𝑗 ∈ R. But in any case,
this is not strictly necessary. After all, preparing 𝑒−𝑖𝐻𝑡 |𝜓⟩ is not a full algorithm, but a possible

intermediate step, after which measurements must be performed to extract the desired information.

For example, one might be interested in estimated computing the dynamics of observables via

⟨𝑂 (𝑡)⟩ = ⟨𝜓𝑡 | 𝑂 |𝜓𝑡⟩ .

(3.3)

This suggests the possibility of summing not operations, but classical data coming from measure-

ment schemes. This amounts to a standard Richardson extrapolation of the expectation values [44].

Let 𝑠 = 1/𝑟 be the "normalized Trotter step." For a given product formula P in 𝑟 steps, the

expectation value

⟨𝑂 𝑠 (𝑡)⟩ := ⟨𝜓0| P†(𝑠𝑡)1/𝑠𝑂P (𝑠𝑡)1/𝑠 |𝜓0⟩

(3.4)

is a real-valued, smooth function 𝑓 (𝑠). By estimating 𝑓 (𝑠 𝑗 ) for various 𝑠 𝑗 , an extrapolation can

be performed to the desired 𝑠 = 0. What is attractive about this scheme is that no additional

quantum resources, beyond the product formulas simulation and expectation value measurements,

are needed.

37

There are several directions in which these ideas can be extended. First, the function 𝑓 (𝑠)

could represent a variety of kinds of measurements obtained from product simulations, beyond

expectation values. For example, they could represent eigenvalue estimates from phase estimation.

Second, we might consider other means of estimating 𝑓 (0) besides Richardson extrapolation. One

possibility is to produce a uniform approximation to 𝑓 in a neighborhood of 𝑠 = 0. Function

approximation is an old science, and many numerical techniques are available for the task. Modern

machine learning approaches such as neural networks might achieve excellent approximation for 𝑓 ,

but finding the simplest effective tool for the task is desirable, and moreover it is difficult to provide

rigorous guarantees for machine learning.

3.2 Polynomial Interpolation

Among the collection of function approximation methods available, we choose one that is

simple to implement and easy to analyze: polynomial interpolation. Essentially, our goal is to use

interpolation to "extrapolate" the Trotter step size 𝑠 to the ideal of 𝑠 = 0.1 There are many quantities

that we could be interested in extrapolating. For this thesis, we will be primarily concerned with

expectation value estimation.

⟨𝑂 𝑠 (𝑡)⟩ = Tr 𝜌𝑂 𝑠 (𝑡)

𝑂 𝑠 (𝑡) : = ˜𝑈𝑠 (𝑡)†𝑂 ˜𝑈𝑠 (𝑡)

(3.5)

For purposes of analysis, we’ll assume these expectation values are estimated on a quantum

computer using one of the Suzuki-Trotter (ST) formulas of equation (2.28). However, in principle

our approach should work for any product formula simulation, not just ST.

While the interpolation is classical and independent of the method in which the data is generated,

we will assume a quantum simulation is used when considering the computational cost. We assume

all quantum operations are executed perfectly, including the exponentials exp(−𝑖𝐻 𝑗 𝑡) for simulation.

This is not to say that the interpolation method could not be applied to noisy quantum systems,

but rather that our cost analysis does not account for it. Consequently, the only sources of error

1We occasionally interchange between the terminology "extra-" and "interpolation." We view our method as an

extrapolation beyond the data using a numerical technique commonly known as polynomial interpolation.

38

considered are the interpolation error and error in the calculation of the data points (e.g.

the

Hamiltonian energies or expectation values at various points 𝑠𝑖). Error in the data points may arise

from hardware noise, but even in its absence, a measurement protocol such as phase estimation

induces a systematic error that cannot be removed.

Without further ado, we now describe the interpolation framework. Let 𝑓 ∈ 𝐶∞( [−𝑎, 𝑎]) be

a smooth, real-valued function of a single variable 𝑠 ∈ [−𝑎, 𝑎] and suppose we have calculated 𝑓

(perfectly) for 𝑛 distinct points 𝑠1, 𝑠2 . . . 𝑠𝑛 ∈ [−𝑎, 𝑎]. That is, we have data in the form of a set
of tuples 𝐷 = {(𝑠𝑖, 𝑓𝑖)}𝑛
𝑖=1, where 𝑓𝑖 = 𝑓 (𝑠𝑖). Let 𝑃𝑛−1 𝑓 be the unique (𝑛 − 1)-degree polynomial
interpolating 𝐷, i.e. 𝑃 𝑓𝑛−1(𝑠𝑖) = 𝑓𝑖 for each 𝑖 = 1, . . . , 𝑛. For any 𝑠 ∈ [−𝑎, 𝑎], standard results in

polynomial interpolation [107] tell us that the signed error is given by

𝐸𝑛−1(𝑠) := 𝑓 (𝑠) − 𝑃𝑛−1 𝑓 (𝑠) =

𝑓 (𝑛) (𝜉)
𝑛!

𝜔𝑛 (𝑠)

(3.6)

for some 𝜉 ∈ 𝐼𝑠, where 𝐼𝑠 ⊂ [−𝑎, 𝑎] is the smallest interval containing 𝑠 and the interpolation

points {𝑠𝑖}. Throughout this work, superscripts such as in 𝑓 (𝑛) will refer to 𝑛th-order derivatives.

The 𝑛th degree nodal polynomial 𝜔𝑛 (𝑠) is defined as the unique monic polynomial with zeros at

the interpolation points.

𝜔𝑛 (𝑠) :=

𝑛
(cid:214)

(𝑠 − 𝑠𝑖)

(3.7)

𝑖=1
Our estimate for 𝑓 (0) is 𝑃𝑛−1 𝑓 (0). Since we are interested in 𝑠 = 0, 𝜔𝑛 becomes a (signed)

product of the interpolation points. We can bound the interpolation error 𝐸𝑛 (0) in a way that is

independent of the precise value of 𝜉 (which is unknown and difficult to find) by maximizing over

𝜉 ∈ 𝐼𝑠.

|𝐸𝑛−1(0)| ≤ max
𝑠∈𝐼𝑠

| 𝑓 (𝑛) (𝑠)|
𝑛!

𝑛
(cid:214)

𝑖=1

|𝑠𝑖 |

(3.8)

Much of the technical work in this dissertation involves finding suitable bounds on the size of the

derivatives 𝑓 (𝑛). In particular, in the expectation values of equation (3.5), 𝑓 (𝑠) = ⟨𝑂 𝑠 (𝑡)⟩.

For reasons which we discuss in the following Section, we choose the Chebyshev nodes on

[−𝑎, 𝑎] as our interpolation points.

𝑠𝑖 = 𝑎 cos

(cid:18) 2𝑖 − 1
2𝑛

(cid:19)

𝜋

39

(3.9)

This allows us to specialize our interpolation error in the manner described in the following lemma.

Lemma 3.2.1. Let 𝑠𝑖, 𝑖 = 1, 2, . . . , 𝑛 be the collection of Chebyshev interpolation points on the

interval [−𝑎, 𝑎]. In the notation above, we have

|𝐸𝑛−1(0)| < max
𝑠∈[−𝑎,𝑎]

| 𝑓 (𝑛) (𝑠)|

(cid:17) 𝑛

.

(cid:16) 𝑎
2𝑛

Proof. For 𝑛 odd, 𝑠 = 0 is one of the interpolation points, so the error is zero and the bound holds

automatically. Hereafter, we only consider 𝑛 even (which will be the case of practical interest).

Using the generic bound (3.8) with the Chebyshev nodes,

|𝐸𝑛−1(0)| ≤ max
𝜉∈[−𝑎,𝑎]

| 𝑓 (𝑛) (𝜉)|

𝑎𝑛

1
𝑛!

𝑛
(cid:214)

𝑖=1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

cos

(cid:18) 2𝑖 − 1
2𝑛

𝜋

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

.

(3.10)

To obtain the lemma, we just need to appropriately bound the product of cosines. Since 𝑛 is even,

𝑛 = 2𝑚 for some 𝑚 ∈ Z+. Moreover, we have a reflectional symmetry about 𝑚, in the sense that

(cid:12)
(cid:12)
cos
(cid:12)
(cid:12)

(cid:18) 2𝑖 − 1
2𝑛

𝜋

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

=

(cid:12)
(cid:12)
(cid:12)
(cid:12)

cos

(cid:18) 2(𝑛 − 𝑖 + 1) − 1
2𝑛

𝜋

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

.

Hence, we only need to take the product over 𝑖 = 1, . . . , 𝑚 and square it.

𝑛
(cid:214)

𝑖=1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

cos

(cid:18) 2𝑖 − 1
2𝑛

𝜋

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

=

(cid:32) 𝑚
(cid:214)

𝑖=1

(cid:19) (cid:33) 2

cos

(cid:18) 2𝑖 − 1
4𝑚

𝜋

To proceed further, let’s reindex the remaining product by 𝑖 → 𝑚 − 𝑖 + 1. This gives

𝑚
(cid:214)

𝑖=1

cos

(cid:18) 2𝑖 − 1
4𝑚

(cid:19)

𝜋

=

=

<

𝑚
(cid:214)

𝑖=1
𝑚
(cid:214)

𝑖=1
𝑚
(cid:214)

𝑖=1

(cid:19)

𝜋

2𝑖 − 1
4𝑚
(cid:19)

(cid:18) 𝜋

cos

sin

−

2
(cid:18) 2𝑖 − 1
4𝑚

2𝑖 − 1
4𝑚

(3.11)

(3.12)

(3.13)

where we used the fact that sin(𝑥) < 𝑥 for all 𝑥 > 0. Factoring out the denominator from the

product, the remaining terms become a double factorial.

𝑚
(cid:214)

𝑖=1

2𝑖 − 1
4𝑚

=

(2𝑚 − 1)!!
(4𝑚)𝑚

40

(3.14)

The double factorial can be bounded as

(2𝑚 − 1)!!2 < (2𝑚 − 1)!!(2𝑚)!! = 2𝑚! ,

(3.15)

so that (2𝑚−1)!! < √︁(2𝑚)!. Returning to the original product of equation (3.12), and reintroducing

𝑛 = 2𝑚, the resulting bound is

𝑛
(cid:214)

𝑖=1

(cid:12)
(cid:12)
(cid:12)
(cid:12)

cos

(cid:18) 2𝑖 − 1
2𝑛

𝜋

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

<

(cid:33) 2

(cid:32) √

𝑛!
(2𝑛)𝑛/2

=

𝑛!
(2𝑛)𝑛

.

(3.16)

Reinserting this result into the last line of equation (3.10) gives the bound stated in the lemma. □

Though Chebyshev interpolation enjoys nice mathematical properties, it presents a challenge

for Trotter simulation because of the need for noninteger time steps in equation (3.44). In the face

of this obstacle, there are several options one could take: rounding to integer time steps, or perform

fractional queries using, say, the Quantum Singular Value Transformation (QSVT).

First, consider rounding to integer time steps, i.e., gathering data at the nearest reciprocal integer

1/ ˜𝑟 to the Chebyshev node 𝑠. For symmetrical interval [−𝑎, 𝑎], the rounding error |𝑠 − 1/ ˜𝑟 | goes

as 𝑂 (𝑎2) as 𝑎 → 0. From here, one could either (a) take the estimate for 𝑓 (1/ ˜𝑟) as the estimate

for 𝑓 (𝑠), accruing some error in the process, or (b) perform the interpolation at the approximate

Chebyshev nodes given by the collection of points 1/𝑟𝑖. Unfortunately, for our purposes, option (a)

leads to unacceptable errors of order 𝑂 (𝑎) in the data, eliminating accuracy gains. As for option

(b), it is possible to use robustness results on Chebyshev interpolation [132] to argue that almost-

Chebyshev nodes should be almost as well-conditioned. Again, however, we find that our scaling

of the number of nodes is such that the node displacements must be quite small, leading again to

poorer scaling. Because of this, for most of this work we choose to invoke access to fractional

queries using the QSVT [54]. While fractional queries increase the overhead compared to Trotter

alone, this overhead is a constant. In practice, it may be that taking approximate Chebyshev nodes

is perfectly acceptable and stable, and if so would likely be the preferred method.

3.3 Stability Analysis

Polynomial interpolation is a valuable numerical tool, but some implementations can lead to

numerical instability [36]. However, the situation is not as bad as often presented in textbooks [126].

41

While linear algebraic approaches involving Vandermonde matrices suffer instability for high degree

polynomials [51], methods such as barycentric formulas are provably stable with respect to floating

point arithmetic [68].

A particularly important consideration is the choice of interpolation nodes. It is well known

that equally spaced nodes can lead to the Runge phenomenon: rapid oscillations near the ends of

the interval that grow with polynomial degree [107]. These oscillations can be overcome with a

superior choice of nodes, such as the zeros of the Chebyshev polynomials. Interpolations done with

this set of nodes are guaranteed to converge to functions that are Lipschitz continuous as 𝑛 → ∞.

Moreover, they are well-conditioned in the sense of small errors in the data values. Finally, because

they anti-cluster around 𝑠 = 0, they are relatively cheap to compute with Trotter formulas. In this

work, we will always interpolate at the 𝑛th-degree Chebyshev nodes, or approximations thereof, on

a symmetric interval [−𝑎, 𝑎] about the origin, defined in (3.9). We choose even 𝑛 so as to avoid the

origin (which has infinite cost to compute), and also utilize the reflectional symmetry of 𝑓 (𝑠).

To compute the interpolant 𝑃𝑛−1 𝑓

linear algebraically, we overcome the limitations of the

standard Vandermonde approach by expanding in terms of orthonormal Chebyshev polynomials

rather than monomials 𝑥 𝑗 .

Here, 𝑝 𝑗 is defined by

𝑃𝑛−1 𝑓 (𝑠) =

𝑛−1
∑︁

𝑗=0

𝑐 𝑗 𝑝 𝑗 (𝑠)

𝑝 𝑗 (𝑠) :=

√︃ 1

𝑛𝑇0(𝑠),

√︃ 2

𝑛𝑇𝑗 (𝑠),

𝑗 = 0

𝑗 = 1, 2, . . .





where 𝑇𝑗 is the standard 𝑗th Chebyshev polynomial.

𝑇𝑗 (𝑥) := cos( 𝑗 cos−1 𝑥)

By orthonormality, we are referring to the condition [93]

𝑛
∑︁

𝑘=1

𝑝𝑖 (𝑠𝑘 ) 𝑝 𝑗 (𝑠𝑘 ) = 𝛿𝑖 𝑗

42

(3.17)

(3.18)

(3.19)

(3.20)

for all 0 ≤ 𝑖, 𝑗 < 𝑛, with 𝑠𝑘 being the zeros of 𝑇𝑛 given in (3.9). This immediately implies the

matrix

V :=

𝑝0(𝑠1) 𝑝1(𝑠1)

. . . 𝑝𝑛−1(𝑠1)

𝑝0(𝑠2) 𝑝1(𝑠2)

...

...

. . . 𝑝𝑛−1(𝑠2)
. . .

...

𝑝0(𝑠𝑛) 𝑝1(𝑠𝑛)

. . . 𝑝𝑛−1(𝑠𝑛)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(3.21)

is orthogonal, and therefore has condition number 𝜅(V) := ∥V∥∥V−1∥ equal to one. This is the

source of well-conditioning in our approach. The coefficients 𝑐 = (𝑐0, 𝑐1, . . . , 𝑐𝑛−1) in equa-

tion (3.17) satisfy

𝑦 = V𝑐

(3.22)

for the vector of values 𝑦 = ( 𝑓 (𝑠1), 𝑓 (𝑠2), . . . , 𝑓 (𝑠𝑛)), since 𝑃𝑛−1 𝑓
𝑐 = V𝑇 𝑦 gives the vector of coefficients.

is an interpolant. Hence,

We now develop our argument for well-conditioning. Unless otherwise subscripted, all loga-

rithms are natural.

Lemma 3.3.1. Let 𝑠1, 𝑠2, . . . , 𝑠𝑛 be the standard Chebyshev nodes on [−𝑎, 𝑎] (3.9) with 𝑛 even.

Then the nodes satisfy

𝑛
∑︁

𝑘=1

1
|𝑠𝑘 |

≤

4𝑛
𝜋𝑎

(𝛾 + log(2𝑛 + 2)) ,

where 𝛾 ≈ 0.577 is the Euler-Mascheroni constant.

Proof. We focus on the case 𝑎 = 1, since the general result follows by a simple rescaling. Because

sine and cosine are phase shifted by 𝜋/2,

𝑛
∑︁

𝑘=1

1
|𝑠𝑘 |

=

𝑛
∑︁

𝑘=1

1
(cid:16) 2𝑘−1
2𝑛 𝜋

(cid:17)(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)cos

=

𝑛
∑︁

𝑘=1

1
(cid:16) 𝑛−2𝑘+1
2𝑛 𝜋

.

(cid:17)(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)sin

Taking advantage of the symmetry about 𝑠 = 0,

𝑛
∑︁

𝑘=1

1
(cid:16) 𝑛−2𝑘+1
2𝑛 𝜋

(cid:17)(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)sin

𝑛/2
∑︁

= 2

𝑘=1

sin

1
(cid:16) 2𝑘−1
2𝑛 𝜋

.

(cid:17)

43

(3.23)

(3.24)

Next, we use the lower bound

sin 𝑥 ≥ 𝑥/2

(0 ≤ 𝑥 ≤ 𝜋/2)

(3.25)

in order to bound the terms of the sum.

𝑛/2
∑︁

2

𝑘=1

sin

8𝑛
𝜋

(cid:17) ≤

1
(cid:16) 2𝑘−1
2𝑛 𝜋

8𝑛
𝜋

=

𝑛/2
∑︁

𝑘=1
(cid:18)

1
2𝑘 − 1

𝐻𝑛 −

(cid:19)

𝐻𝑛/2

1
2

(3.26)

Here, 𝐻𝑛 denotes the 𝑛th harmonic number. From the relation 𝐻𝑛 = 𝛾 + 𝜓(𝑛 + 1), where 𝜓 is the

digamma function,

𝐻𝑛 −

1
2

𝐻𝑛/2 = 𝛾/2 + 𝜓(𝑛 + 1) −

𝜓(𝑛/2 + 1).

1
2

(3.27)

Moreover, since 𝜓(𝑥) ∈ (log(𝑥 − 1/2), log(𝑥)) for any 𝑥 > 1/2, this is upper bounded by

𝐻𝑛 −

1
2

𝐻𝑛/2 < 𝛾/2 + log(𝑛 + 1) −

1
2

log(

𝑛 + 1
2

) =

𝛾 + log 2
2

+

log(𝑛 + 1)
2

.

(3.28)

Reinserting this into (3.26), one obtains the bound

𝑛
∑︁

𝑘=1

1
|𝑠𝑘 |

≤

4𝑛
𝜋

(𝛾 + log(2𝑛 + 2)) .

The general lemma follows from a rescaling by 1/𝑎.

(3.29)

□

Observe that 1/|𝑠𝑘 | is essentially the number of Trotter steps to compute the 𝑘th interpolation

point. Thus, Lemma 3.3.1 amounts to a bound on the total number of Trotter steps, and we see this

grows as 𝑂 (𝑎−1𝑛 log 𝑛).

Lemma 3.3.2. Let 𝑝(𝑠) = ( 𝑝0(𝑠), 𝑝1(𝑠), . . . , 𝑝𝑛−1(𝑠)) be a vector of (normalized) Chebyshev

polynomials on [−𝑎, 𝑎]. Then,

∥V𝑝(0) ∥1 < 2

𝜋 log (𝑛 + 1) + 1

where ∥ · ∥1 denotes the vector 1-norm.

44

Proof. Let 𝑑 (𝑠) = V𝑝(𝑠). For each 𝑘 = 1, 2, . . . 𝑛 we have

𝑑𝑘 (𝑠) =

𝑛−1
∑︁

𝑗=0

V𝑘 𝑗 𝑝 𝑗 (𝑠) =

𝑛−1
∑︁

𝑝 𝑗 (𝑠𝑘 ) 𝑝 𝑗 (𝑠)

=

1
𝑛

+

2
𝑛

(cid:18)

𝑗

cos

𝑛−1
∑︁

𝑗=1

𝑗=0
(cid:18) 2𝑘 − 1
2𝑛

(cid:19)(cid:19)

𝜋

cos( 𝑗 cos−1(𝑠)).

At 𝑠 = 0, cos( 𝑗 cos−1(0)) = cos( 𝑗 𝜋/2), which is zero for odd 𝑗. Hence,

𝑑𝑘 (0) =

=

1
𝑛

1
𝑛

+

+

2
𝑛

2
𝑛

𝑛−2
∑︁

𝑗=2,even
𝑛/2−1
∑︁

𝑗 ′=1

cos

(cid:18)

𝑗

(cid:18) 2𝑘 − 1
2𝑛

(cid:19)(cid:19)

𝜋

(−1) 𝑗/2

(−1) 𝑗 ′

(cid:18)

𝜋 𝑗 ′ 2𝑘 − 1
𝑛

(cid:19)

.

cos

The sum can be evaluated exactly (we used Mathematica), yielding

𝑑𝑘 (0) =

=

=

1
𝑛

1
𝑛

1
𝑛

(cid:32)

(cid:18)

−

−

2
𝑛

1
𝑛

1 − cos((𝑘 + 𝑛/2)𝜋) tan(𝜋 2𝑘−1
2𝑛 )
2

(cid:33)

1 − (−1) 𝑘+𝑛/2 tan(

2𝑘 − 1
2𝑛

(cid:19)

𝜋)

(−1) 𝑘+𝑛/2 tan

(cid:18) 2𝑘 − 1
2𝑛

(cid:19)

𝜋

.

With coefficients in hand, we now compute the one norm of 𝑑 (0).
(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:18) 2𝑘 − 1
2𝑛

∥𝑑 (0)∥1 =

𝑛
∑︁

tan

1
𝑛

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

𝑘=1
We have a reflectional symmetry about 𝑘 → 𝑛 − 𝑘 + 1, allowing us to cut the sum in half and remove

the absolute value sign.

∥𝑑 (0) ∥1 =

2
𝑛

𝑛/2
∑︁

tan

𝑘=1
𝑚
∑︁

tan

(cid:19)

𝜋

(cid:18) 2𝑘 − 1
2𝑛
(cid:18) 2𝑘 − 1
2𝑚

2

=

1
𝑚

(cid:19)

𝜋

(3.36)

𝑘=1
Here, 𝑚 ≡ 𝑛/2. We observe that the sum increases as 𝑘 approaches 𝑚 due to the first order pole at

𝜋/2. We can upper bound tan(𝑥), and therefore the sum above, as follows.
(cid:19)

1
𝑚

𝑚
∑︁

𝑘=1

tan

(cid:18) 2𝑘 − 1
2𝑚

𝜋

2

𝑚
∑︁

≤

1
𝑚

(cid:17) =

4
𝜋

1
(cid:16) 2𝑘−1
2𝑚

𝑘=1

𝜋

2 − 𝜋

2

1
2(𝑚 − 𝑘) + 1

(3.37)

1
2 𝑗 − 1

𝑚
∑︁

𝑘=1
𝑚
∑︁

𝑗=1

=

4
𝜋

45

In the last line, we reindexed by 𝑗 = 𝑚 − 𝑘 + 1. Borrowing the reasoning from the prior lemma,

4
𝜋

𝑚
∑︁

𝑗=1

1
2 𝑗 − 1

< 2
𝜋

(𝛾 + log(2𝑛 + 2)).

Tracing back, this is an upper bound on ∥𝑑 (0) ∥1. Hence,

∥𝑑 (0)∥1 < 2

𝜋 log(𝑛 + 1) +

2(𝛾 + log(2))
𝜋

< 2

𝜋 log(𝑛 + 1) + 1.

(3.38)

(3.39)

□

The benefit of well-conditioning comes from relaxing the need to have exquisitely precise data

to achieve good interpolations. This property is captured by the following Theorem.

Theorem 3.3.3. Let 𝑦 = ( 𝑓 (𝑠1), 𝑓 (𝑠2), . . . , 𝑓 (𝑠𝑛))𝑇 , and let ˜𝑦 ∈ R𝑛 be an approximation of 𝑦 in
𝜋 log(𝑛 + 1) + 1) with probability at least
the sense that, for all 1 ≤ 𝑗 ≤ 𝑛, | 𝑓 (𝑠 𝑗 ) − ˜𝑦 𝑗 | ≤ 𝜖/( 2
1 − 𝛿/𝑛. Let 𝑝(𝑠) = ( 𝑝0(𝑠), . . . , 𝑝𝑛−1(𝑠))𝑇 be the vector of orthonormal Chebyshev polynomials.
Then ˜𝑦𝑇 V𝑝(𝑠) is an estimate of the interpolant 𝑃𝑛−1 𝑓 (𝑠) at 𝑠 = 0 to precision

|𝑃𝑛−1 𝑓 (0) − ˜𝑦𝑇 V𝑝(0)| ≤ 𝜖

with probability at least 1 − 𝛿.

Proof. First, observe that 𝑃𝑛−1 𝑓 (𝑠) = 𝑝(𝑠)𝑇 𝑐 = 𝑝(𝑠)𝑇 V𝑇 𝑦 by the discussion surrounding (3.22).

Hence,

By Hölder’s inequality,

|𝑃𝑛−1 𝑓 (0) − 𝑝(0)𝑇 V𝑇 ˜𝑦| = |(V𝑝(0))𝑇 (𝑦 − ˜𝑦)|.

|(V𝑝(0))𝑇 (𝑦 − ˜𝑦)| ≤ ∥V𝑝(0) ∥1∥𝑦 − ˜𝑦∥∞.

From Lemma 3.3.2, and from the assumptions on the distance between 𝑦 and ˜𝑦,

∥V𝑝(𝑠)∥1∥𝑦 − ˜𝑦∥∞ ≤

(cid:18) 2
𝜋 log(𝑛 + 1) + 1

(cid:19)

𝜖
𝜋 log(𝑛 + 1) + 1
2

= 𝜖

(3.40)

(3.41)

(3.42)

with probability Pr = (1 − 𝛿/𝑛)𝑛. In fact, since the probability of each component ˜𝑦 exceeding

the specified distance is 𝛿/𝑛, by the union bound the total probability of at least one component

46

exceeding this distance is less than 𝑛 × (𝛿/𝑛) = 𝛿. Thus, the inequality is satisfied with probability

Pr ≥ 1 − 𝛿. This completes the proof.

□

Theorem 3.3.3 is what suggests that our interpolation approach may have the potential to

achieve accuracy improvements without increasing costs compared to standard Trotter.

It tells

us that the error in Trotter data can be as large as the error of the final estimate up to a factor

which is logarithmically small in the number of interpolation points, and therefore these data ˜𝑦𝑖

can be computed "cheaply enough." Thus, Theorem 3.3.3 is plays an important role in the proofs

of Lemma 3.5.1, presented in Section 3.5.

3.4 The Effective Hamiltonian

Trotter formulas approximate 𝑈 (𝑡) only in a neighborhood around 𝑡 = 0; thus the standard

procedure for product formula simulations, as described in Section 2.7.1, is to divide the simulation

interval [0, 𝑡] into 𝑟 subintervals, such that each interval is sufficiently small that the Trotter

approximation is valid. For the simple case of a uniform mesh of 𝑟 subintervals, this becomes

𝑆2𝑘 (𝑡/𝑟)𝑟 = 𝑈 (𝑡) + 𝑂 (𝑡2𝑘+1/𝑟 2𝑘 )

(3.43)

where big 𝑂 is understood as taking 𝑟 large. However, it is simpler for our subsequent analysis to

consider 𝑠 = 1/𝑟 as a "dimensionless step size," and instead think about 𝑠 → 0. In terms of 𝑠, we

define

˜𝑈𝑠 (𝑡) := 𝑆2𝑘 (𝑠𝑡)1/𝑠

(3.44)

as the approximate evolution operator for 𝑠 ≠ 0. The discontinuity at 𝑠 = 0 in (3.44) may be filled

by the exact evolution ˜𝑈0(𝑡) := 𝑈 (𝑡). Though we defined 𝑠 as a reciprocal integer, definition (3.44)

suggests an extension to allow 𝑠 to be real-valued. In fact, the resulting function ˜𝑈𝑠 is smooth on

a neighborhood of 𝑠 = 0, a fact that will allow us to precisely characterize the interpolation error.

For our purposes, we will only consider |𝑠| ≤ 1. When 1/𝑠 is not an integer, we may implement ˜𝑈𝑠

using fractional queries [54] by splitting 1/𝑠 into integer and fractional parts.

1/𝑠 = 𝑟 + 𝑓

47

(3.45)

Here, 𝑟 = rnd(1/𝑠) ∈ Z is 1/𝑠 rounded to the nearest integer, and 𝑓 ∈ [−1/2, 1/2]. Finally, we

note that ˜𝑈𝑠 is an even function of 𝑠, which we will make use of to cut the number of interpolation

points in half by reflecting across 𝑠 = 0.

Prior work has demonstrated the value of considering the effective Trotter Hamiltonian in the

analysis of Trotter formulas [139]. This approach is also helps us calculate high order derivatives

of ˜𝑈𝑠 as needed for our error bounds. We define an effective Hamiltonian

so that

˜𝐻𝑠 :=

𝑖
𝑠𝑡 log 𝑆2𝑘 (𝑠𝑡)

˜𝑈𝑠 (𝑡) = 𝑒−𝑖 ˜𝐻𝑠𝑡 .

(3.46)

(3.47)

Note that ˜𝐻𝑠 depends on 𝑡 as well, though this dependence will be left implicit. For the purposes

of bounding the interpolation error, we require a bound on the norm of ˜𝐻𝑠. This is supplied by the

following lemma.

Lemma 3.4.1. In the notation introduced above, let 𝑠 be chosen such that

𝑘 (5/3) 𝑘 𝑚 max
𝑙∈[1,𝑚]

∥𝐻𝑙 ∥|𝑠|𝑡 ≤ 𝜋/20.

Then the following bound on the derivatives of ˜𝐻𝑠 with respect to 𝑠 holds.

∥𝜕𝑛

𝑠 ˜𝐻𝑠 ∥ ≤ 2𝑡−1𝑛𝑛 (𝑒2𝑘 (5/3) 𝑘 𝑚 max
𝑙∈[1,𝑚]

∥𝐻𝑙 ∥𝑡)𝑛+1

Note that our bounds are uniformly worse for larger 𝑘, i.e., higher order ST formulas. Assuming

that this is not an artifact of our mathematical treatment, this suggests low order formulas are

unconditionally preferred over high order ones for interpolation. Numerical studies could help

determine the true impact of higher order formulas on the interpolation procedure.

We conclude this section with the proof of the above Lemma, which will be essential to our

subsequent error analysis. The upper bound will prove useful because, as we will see, the error

in polynomial interpolation can be expressed using a formula akin to the Taylor remainder, which

involves a high-order derivative.

48

Proof of Lemma 3.4.1. Recall the definition of the effective Hamiltonian (3.46), defined for 𝑠 ∈

R \ {0} and for 𝑠 = 0 by ˜𝐻0 := lim𝑠→0 ˜𝐻𝑠 = 𝐻. We will understand log 𝑆2𝑘 (𝑠𝑡) through a power

series expansion about the identity.

log 𝑆2𝑘 (𝑠𝑡) =

∞
∑︁

𝑗=0

(−1) 𝑗
𝑗 + 1

(𝑆2𝑘 (𝑠𝑡) − 𝐼) 𝑗+1

This series converges precisely when

∥𝑆2𝑘 (𝑠𝑡) − 𝐼 ∥ ≤ 1.

(3.48)

(3.49)

Using the fundamental theorem of calculus, we can derive a suitable condition for convergence as

a neighborhood about 𝑠 = 0. The condition above implies

which is satisfied provided

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∫ 𝑠𝑡

0

𝑆2𝑘 (𝑥)𝑑𝑥

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤ 1

|𝑠𝑡| max
𝑥∈[0,𝑠𝑡]

(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑑
𝑑𝑥

𝑆2𝑘 (𝑥)

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤ 1.

(3.50)

(3.51)

Writing out 𝑆2𝑘 (𝑥) = (cid:206)𝑁𝑘

𝑙=1 exp(−𝑖𝐻 𝑗𝑙 𝜏𝑙𝑥) where 𝐻 𝑗𝑙 is some Hamiltonian piece 𝐻 𝑗 indexed by 𝑙,

the derivative can be upper bounded as

(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑑
𝑑𝑥

max
𝑥∈[0,𝑠𝑡]

𝑆2𝑘 (𝑥)

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤

𝑁𝑘∑︁

𝑙=1

∥𝐻 𝑗𝑙 ∥|𝜏𝑙 |

≤ max
𝑗

∥𝐻 𝑗 ∥∥𝜏∥1

(3.52)

where 𝜏 = (𝜏𝑙) 𝑁𝑘

𝑙=1 is the vector of ST coefficients, and in going to the second line we used a Hölder

inequality. We have ∥𝜏𝑙 ∥1 ≤ 𝑁𝑘 max𝑙 |𝜏𝑙 |, and from Appendix A of [137] we have

|𝜏𝑙 | ≤ 2𝑘/3𝑘 .

max
𝑙

Thus, the requirement for convergence of the logarithm becomes

4
3

𝑘 (5/3) 𝑘−1𝑚|𝑠𝑡| max

𝑗

∥𝐻 𝑗 ∥ ≤ 1

where we used the expression 𝑁𝑘 = (2𝑚)5𝑘−1 for the number of ST exponentials.

(3.53)

(3.54)

49

We now assume 𝑠 is within the symmetric interval defined by (3.54), such that (3.48) is

convergent. Since log 𝑆2𝑘 (0) = 0, 𝑠 = 0 is a zero of order at least one. We want to absorb the

diverging 1/𝑠 term and better understanding the leading dependence in 𝑠. To facilitate this, we

write

where we defined

˜𝐻𝑠 = −

1
𝑖𝑡

∞
∑︁

𝑗=0

(−1) 𝑗
𝑗 + 1

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1

Δ𝑆2𝑘 (𝑠𝑡) :=

𝑆2𝑘 (𝑠𝑡) − 𝐼
𝑠

.

Note that Δ𝑆2𝑘 is analytic in 𝑠, and is a finite difference around 𝑠 = 0, such that

Δ𝑆2𝑘 (𝑠𝑡) = −𝑖𝐻𝑡.

lim
𝑠→0

(3.55)

(3.56)

(3.57)

Through the series expansion (3.55) we may bound derivatives of ˜𝐻𝑠 via bounds on derivatives of

Δ𝑆2𝑘 . We first obtain a power series of Δ𝑆2𝑘 by Taylor expanding every term in the product formula

𝑆2𝑘 . Regrouping in powers of 𝑠𝑡, the result is

Δ𝑆2𝑘 (𝑠𝑡) =

∞
∑︁

𝑗=1

𝑠 𝑗−1(−𝑖𝑡) 𝑗
𝑗!

(cid:18)

∑︁

𝐽

𝑗
𝑗1 . . . 𝑗𝑁𝑘

(cid:19) 𝑁𝑘(cid:214)

𝑙=1

(𝐻𝑙𝜏𝑙) 𝑗𝑙

(3.58)

where the parenthetical symbol is the multinomial coefficient, and the sum (cid:205)𝐽 is over all values of
𝐽 = ( 𝑗1, . . . , 𝑗𝑁𝑘 ) such that (cid:205)𝑘 𝑗𝑘 = 𝑗. The derivatives with respect to 𝑠 are now easy to compute.

Using the fact that

𝑠 𝑠 𝑗−1 =
𝜕𝑛

( 𝑗 − 1)!
( 𝑗 − 1 − 𝑛)!

𝑠 𝑗−𝑛−1

for 𝑗 > 𝑛 (and zero otherwise), we have

𝜕𝑛
𝑠 Δ𝑆2𝑘 (𝑠𝑡) =

∥𝜕𝑛

𝑠 Δ𝑆2𝑘 (𝑠𝑡)∥ ≤

∞
∑︁

𝑗=𝑛+1
∞
∑︁

𝑗=𝑛+1

𝑠 𝑗−𝑛−1(−𝑖𝑡) 𝑗
𝑗!

( 𝑗 − 1)!
( 𝑗 − 1 − 𝑛)!

(cid:18)

∑︁

𝐽

𝑗
𝑗1 . . . 𝑗𝑁𝑘

(cid:19) 𝑁𝑘(cid:214)

𝑙=1

(𝐻𝑙𝜏𝑙) 𝑗𝑙

𝑡 𝑗
( 𝑗 − 𝑛 − 1)!

𝑠 𝑗−𝑛−1(𝜏max𝑁𝑘 Λ) 𝑗

(3.59)

(3.60)

50

where Λ := max 𝑗 ∥𝐻 𝑗 ∥ and 𝜏max = max𝑙 |𝜏𝑙 |. Factoring out powers of 𝑛 + 1 and reindexing, we are

left with the following bound on derivatives of Δ𝑆2𝑘 .

∥𝜕𝑛

𝑠 Δ𝑆2𝑘 (𝑠𝑡) ∥ ≤ (𝜏max𝑁𝑘 Λ𝑡)𝑛+1𝑒𝑠𝜏max𝑁𝑘Λ𝑡

(3.61)

This expression is quite elegant; it is as if we were taking 𝑛 + 1 derivatives of the exponential 𝑒𝑐𝑠

with

𝑐 := 𝜏max𝑁𝑘 Λ𝑡

≤ 𝑘 (5/3) 𝑘 𝑚Λ𝑡.

(3.62)

Factors of 𝑐 will occur frequently in what follows, so we find it convenient to adopt this symbol as

shorthand.

We return to bounding the derivatives of powers of Δ𝑆2𝑘 (𝑠𝑡) as in equation (3.55).

𝜕𝑛
𝑠

(cid:2)Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:3)

We reduce this to the previous case by performing a multinomial expansion.

𝑠 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1 =
𝜕𝑛

(cid:18)

∑︁

𝑁

𝑛
𝑛0 . . . 𝑛 𝑗

(cid:19)

𝑗
(cid:214)

𝑙=0

𝜕𝑛𝑙
𝑠 Δ𝑆2𝑘 (𝑠𝑡)

(3.63)

(3.64)

As usual, the capital letter 𝑁 denotes the set of all nonnegative indices 𝑛0, . . . , 𝑛 𝑗 summing to 𝑛.

Applying the triangle inequality and submultiplicativity, and employing the bound (3.61),

∥𝜕𝑛

𝑠 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1∥ ≤

(cid:18)

∑︁

𝑁

𝑛
𝑛0 . . . 𝑛 𝑗

(cid:19)

𝑗
(cid:214)

∥𝜕𝑛𝑙

𝑠 Δ𝑆2𝑘 (𝑠𝑡) ∥

(cid:19)

(cid:18)

≤

∑︁

𝑛
𝑛0 . . . 𝑛 𝑗
𝑙=0
(cid:18)
= 𝑒( 𝑗+1)𝑐𝑠𝑐𝑛+ 𝑗+1 ∑︁

𝑁

𝑙=0
𝑗
(cid:214)

𝑁

𝑐𝑛𝑙+1𝑒𝑐𝑠

𝑛
𝑛0 . . . 𝑛 𝑗

(cid:19)

,

(3.65)

where we’ve used the sum property of the 𝑛𝑙 where appropriate. The remaining sum over the

multinomial coefficient is given by ( 𝑗 + 1)𝑛. Hence,

∥𝜕𝑛

𝑠 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1∥ ≤ (( 𝑗 + 1)𝑐)𝑛 (𝑐𝑒𝑐𝑠) 𝑗+1.

(3.66)

51

Notice that, when 𝑗 = 0, this is consistent with equation (3.61).

With result (3.66) in hand, we return to the power series (3.55). Differentiating term by term

𝜕𝑛
𝑠 ˜𝐻𝑠 = −

1
𝑖𝑡

∞
∑︁

𝑗=0

(−1) 𝑗
𝑗 + 1

𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

and performing a binomial expansion for each term

𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

=

𝑛
∑︁

𝑞=0

(cid:19)

(cid:18)𝑛
𝑞

𝑠 𝑠 𝑗 (cid:1) (cid:16)
(cid:0)𝜕𝑞

𝑠 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)
𝜕𝑛−𝑞

(3.67)

(3.68)

will allow us to apply our previous results. It will be helpful to consider two cases separately: 𝑗 ≤ 𝑛

and 𝑗 > 𝑛. These regimes are somewhat qualitatively different, since the derivatives of 𝑠 𝑗 may or

may not vanish depending on the number of derivatives. Focusing on the case 𝑗 ≤ 𝑛, we have

𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

=

𝑗
∑︁

𝑞=0

(cid:19)

(cid:18)𝑛
𝑞

𝑗!
( 𝑗 − 𝑞)!

𝑠 𝑗−𝑞 (cid:16)

𝑠 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)
𝜕𝑛−𝑞

.

(3.69)

Note that the sum runs only to 𝑗, not 𝑛. Taking a triangle inequality upper bound using (3.66), we

may upper bound (3.69) as

𝑗
∑︁

𝑞=0

(cid:19)

(cid:18)𝑛
𝑞

𝑗!
( 𝑗 − 𝑞)!

𝑠 𝑗−𝑞 (( 𝑗 + 1)𝑐)𝑛−𝑞 (𝑐𝑒𝑐𝑠) 𝑗+1

= (𝑐𝑒𝑐𝑠) 𝑗+1

𝑗
∑︁

𝑞=0

(cid:19)

(cid:18) 𝑗
𝑞

𝑛!
(𝑛 − 𝑞)!

𝑠 𝑗−𝑞 (( 𝑗 + 1)𝑐)𝑛−𝑞

(3.70)

where we have factored out terms not involving 𝑞 from the sum, and manipulated the factorials for

reasons which will be seen presently. Taking the upper bound 𝑛!/(𝑛 − 𝑞)! < 𝑛𝑞, and factoring out

𝑛 − 𝑗 powers of ( 𝑗 + 1)𝑐, we may upper bound the above expression by

(𝑐𝑒𝑐𝑠) 𝑗+1(( 𝑗 + 1)𝑐)𝑛− 𝑗

𝑗
∑︁

𝑞=0

(cid:19)

(cid:18) 𝑗
𝑞

𝑛𝑞 (( 𝑗 + 1)𝑐𝑠) 𝑗−𝑞

= (𝑐𝑒𝑐𝑠) 𝑗+1(( 𝑗 + 1)𝑐)𝑛− 𝑗 (𝑛 + ( 𝑗 + 1)𝑐𝑠) 𝑗 .

Thus, with some minor polishing, we may express the bound on (3.68) for 𝑗 ≤ 𝑛 as

∥𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

∥ ≤ 𝑒( 𝑗+1)𝑐𝑠𝑐𝑛+1( 𝑗 + 1)𝑛

(cid:18) 𝑛

𝑗 + 1

(cid:19) 𝑗

.

+ 𝑐𝑠

52

(3.71)

(3.72)

Now let’s move on to the 𝑗 > 𝑛 case. Here, there are not enough derivatives to kill off the 𝑠 𝑗

term, so the binomial sum in (3.69) will run from 𝑞 = 0 to 𝑛.

𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

=

𝑛
∑︁

𝑞=0

(cid:19)

(cid:18)𝑛
𝑞

𝑗!
( 𝑗 − 𝑞)!

𝑠 𝑗−𝑞 (cid:16)

𝑠 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)
𝜕𝑛−𝑞

(3.73)

Similar to before, we use the bound (3.66), to obtain

∥𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

∥ ≤

𝑛
∑︁

𝑞=0

(cid:19)

(cid:18)𝑛
𝑞

𝑠 𝑗−𝑞 (( 𝑗 + 1)𝑐)𝑛−𝑞 (𝑐𝑒𝑐𝑠) 𝑗+1

= (𝑐𝑒𝑐𝑠) 𝑗+1𝑠 𝑗−𝑛

(cid:19)

(cid:18)𝑛
𝑞

𝑗!
( 𝑗 − 𝑞)!

(( 𝑗 + 1)𝑐𝑠)𝑛−𝑞.

(3.74)

𝑗!
( 𝑗 − 𝑞)!
𝑛
∑︁

𝑞=0

Taking 𝑗!/( 𝑗 − 𝑞)! < 𝑗 𝑞, a simpler upper bound is given by

(𝑐𝑒𝑐𝑠) 𝑗+1𝑠 𝑗−𝑛

𝑛
∑︁

𝑞=0

(cid:19)

(cid:18)𝑛
𝑞

𝑗 𝑞 (( 𝑗 + 1)𝑐𝑠)𝑛−𝑞 = (𝑐𝑒𝑐𝑠) 𝑗+1𝑠 𝑗−𝑛 ( 𝑗 + ( 𝑗 + 1)𝑐𝑠)𝑛.

(3.75)

With some minor rearrangements, this gives the following upper bound for 𝑗 > 𝑛.

∥𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

∥ ≤ 𝑒( 𝑗+1)𝑐𝑠𝑐𝑛+1( 𝑗 + 1)𝑛 (𝑐𝑠) 𝑗−𝑛

(cid:18)

𝑗
𝑗 + 1

(cid:19) 𝑛

+ 𝑐𝑠

(3.76)

With the bounds (3.72) and (3.76), we can return to bounding 𝜕𝑛

𝑠 ˜𝐻𝑠. Still separating the two

cases 𝑗 ≤ 𝑛 and 𝑗 > 𝑛, we can write

∥𝜕𝑛

𝑠 ˜𝐻𝑠 ∥𝑡 ≤

𝑛
∑︁

𝑗=0

1
𝑗 + 1

= 𝐵𝑙 + 𝐵ℎ

∥𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

∥ +

∞
∑︁

𝑗=𝑛+1

1
𝑗 + 1

∥𝜕𝑛
𝑠

(cid:16)

𝑠 𝑗 Δ𝑆2𝑘 (𝑠𝑡) 𝑗+1(cid:17)

∥

(3.77)

where 𝐵𝑙 and 𝐵ℎ refer to bounds on the "low" and "high" parts of the series. Employing the bounds

from equations (3.72) and (3.76), we have

𝐵𝑙 ≤

𝑛
∑︁

𝑗=0

= 𝑐𝑛+1

1
𝑗 + 1
𝑛
∑︁

𝑒( 𝑗+1)𝑐𝑠𝑐𝑛+1( 𝑗 + 1)𝑛

(cid:18) 𝑛

𝑗 + 1

(cid:19) 𝑗

+ 𝑐𝑠

𝑒( 𝑗+1)𝑐𝑠 ( 𝑗 + 1)𝑛−1

(cid:18)

𝑐𝑠 +

(cid:19) 𝑗

𝑛
𝑗 + 1

(3.78)

𝑗=0

53

and

𝐵ℎ ≤

∞
∑︁

𝑗=𝑛+1

1
𝑗 + 1

𝑒( 𝑗+1)𝑐𝑠𝑐𝑛+1( 𝑗 + 1)𝑛 (𝑐𝑠) 𝑗−𝑛

(cid:18)

𝑗
𝑗 + 1

(cid:19) 𝑛

+ 𝑐𝑠

≤ 𝑐𝑛+1

∞
∑︁

𝑗=𝑛+1

𝑒( 𝑗+1)𝑐𝑠 ( 𝑗 + 1)𝑛−1(𝑐𝑠) 𝑗−𝑛 (1 + 𝑐𝑠)𝑛

(3.79)

= 𝑐𝑛+1 (1 + 𝑐𝑠)𝑛

∞
∑︁

𝑗=𝑛+1

𝑒( 𝑗+1)𝑐𝑠 ( 𝑗 + 1)𝑛−1(𝑐𝑠) 𝑗−𝑛.

Let’s begin by simplifying the bound on 𝐵𝑙. We will at this point make the assumption that

𝑠 is sufficiently small such that 𝑐𝑠 < 1. This will necessarily factor into the cost later. This

simplification yields

𝐵𝑙 ≤ 𝑐𝑛+1

≤ 𝑐𝑛+1

𝑛
∑︁

𝑗=0
𝑛
∑︁

𝑗=0

𝑒 𝑗+1( 𝑗 + 1)𝑛−1

(cid:18)

1 +

(cid:19) 𝑗

𝑛
𝑗 + 1

𝑒 𝑗+1( 𝑗 + 1)𝑛−1𝑒𝑛

≤

1
𝑒

(𝑒2𝑐)𝑛+1

𝑛+1
∑︁

𝑗=1

𝑗 𝑛−1.

The remaining sum can be bounded by (𝑛 + 1)𝑛, hence,

𝐵𝑙 ≤

(𝑒2(𝑛 + 1)𝑐)𝑛+1
𝑒(𝑛 + 1)

≤ (𝑒2𝑐)𝑛+1𝑛𝑛,

(3.80)

(3.81)

where the definition that 00 = 1 handles the edge case. Let’s turn our attention to 𝐵ℎ. We will start

by reindexing so that the series begins at 𝑗 = 0 in (3.79).

𝐵ℎ ≤ 𝑐𝑛+1(1 + 𝑐𝑠)𝑛

∞
∑︁

𝑗=0

𝑒( 𝑗+𝑛+2)𝑐𝑠 ( 𝑗 + 𝑛 + 2)𝑛−1(𝑐𝑠) 𝑗+1

= (𝑐𝑒𝑐𝑠)𝑛+1(1 + 𝑐𝑠)𝑛

∞
∑︁

𝑗=0

(𝑐𝑠𝑒𝑐𝑠) 𝑗+1( 𝑗 + 𝑛 + 2)𝑛−1

The series converges if and only if

𝑐𝑠𝑒𝑐𝑠 < 1.

(3.82)

(3.83)

(3.84)

This condition is slightly stronger than the condition (3.54) that we need for convergence of the

logarithm (3.48), and is equivalent to 𝑐𝑠 < 𝑊 (1) ≈ 0.567, where 𝑊 is the principal brach of the

54

Lambert W function. Returning to (3.82), we have the bound

( 𝑗 + 𝑛 + 2)𝑛−1 = (𝑛 + 2)𝑛−1

(cid:19) 𝑛−1

(cid:18)

1 +

𝑗
𝑛 + 2

≤ (𝑛 + 2)𝑛−1𝑒 𝑗 .

(3.85)

Thus, we have

𝐵ℎ ≤ (𝑐𝑒𝑐𝑠)𝑛+1(1 + 𝑐𝑠)𝑛 (𝑛 + 2)𝑛−1

= (𝑐𝑒𝑐𝑠)𝑛+1(1 + 𝑐𝑠)𝑛 (𝑛 + 2)𝑛−1

(𝑒𝑐𝑠𝑒𝑐𝑠) 𝑗

∞
∑︁

𝑗=0

1
1 − 𝑒𝑐𝑠𝑒𝑐𝑠

.

(3.86)

To be concrete, let’s take 𝑒𝑐𝑠𝑒𝑐𝑠 < 1/2, which is implied by 𝑐𝑠 < 𝜋/20. Coupled with the inequality

in (3.62), this condition can be met provided that

𝑘 (5/3) 𝑘 𝑚Λ𝑠𝑡 ≤ 𝜋/20,

(3.87)

which is exactly the assumption of Lemma 3.4.1. This allows us to upper bound 𝐵ℎ further as

𝐵ℎ ≤ 2𝑒𝜋/20(𝑐𝑠)𝑛+1(3𝑒𝜋/20/2)𝑛 (𝑛 + 2)𝑛−1 ≤ 4(𝑐𝑠)𝑛+1(9/5)𝑛 (𝑛 + 2)𝑛−1.

(3.88)

Since (𝑛 + 2)𝑛−1 ≤ 𝑒2𝑛𝑛/2 (using 00 := 1 for the edge case 𝑛 = 0), we have

Altogether, using 𝑠 ≤ 1

𝐵ℎ ≤ 2𝑒2(𝑐𝑠)𝑛+1 (9/5)𝑛 𝑛𝑛.

∥𝜕𝑛

𝑠 ˜𝐻𝑠 ∥𝑡 ≤ 𝑛𝑛 (𝑒2𝑐)𝑛+1 (cid:16)

1 + 2(9/5𝑒2)𝑛(cid:17)

≤ 2𝑛𝑛 (𝑒2𝑐)𝑛+1.

(3.89)

(3.90)

The final result then follows from substituting for 𝑐 and noting that the duration of each time step

is at most 2𝑘/3𝑘−1 using the results of [138].

□

3.5 Application to Dynamical Observables

We now consider the application of Chebyshev interpolation to estimate expectation values, a

fundamental task in quantum computation. The setting is as follows: given a quantum state 𝜌 and

observable 𝑂, the expectation value is given by ⟨𝑂⟩ = Tr(𝜌𝑂). We evolve our system according

55

to a 2𝑘-th order ST formula ˜𝑈𝑠 given by (3.44). The time evolved expectation values of interest is

captured by the function

𝑓 (𝑠) :=

Tr(𝜌𝑂 𝑠 (𝑡))
∥𝑂 ∥

(3.91)

where 𝑂 𝑠 (𝑡) is given by equation (3.5). We’ve normalized the expectation values by ∥𝑂 ∥ because

the relative error is a useful and natural metric, and also the normalized operators may be block

encoded for amplitude estimation. Alternatively, we simply restrict our attention to normalized

observables with ∥𝑂 ∥ = 1. The interpolation algorithm we propose can be summarized as follows.

1. Given Hamiltonian 𝐻, simulation time 𝑡, and tolerance 𝜖 for the estimate of ⟨𝑂 (𝑡)⟩/∥𝑂 ∥,

choose the appropriate interpolation interval [−𝑎, 𝑎] and an even number 𝑛 of Chebyshev

nodes. We neglect the cost of this step. The error analysis we will perform subsequently will

inform the choices of 𝑎 and 𝑛.

2. Compute estimates ˜𝑦𝑖 of the expectation values ⟨𝑂 𝑠𝑖 (𝑡)⟩ for each 𝑠𝑖 with 𝑖 = 1, . . . , 𝑛/2, to

an accuracy depending on 𝜖 and 𝑛. We will assume this step is done with Iterative Quantum

Amplitude Estimation (IQAE) [56], a recent approach to amplitude estimation that exhibits

low quantum overhead. Our metric of cost is the number of 𝐻 𝑗 exponentials executed on

a quantum circuit, where 𝐻 = (cid:205) 𝑗 𝐻 𝑗 . Note that by symmetry, we need not compute ˜𝑦𝑖 for

𝑖 > 𝑛/2. We have 𝑓 (𝑠𝑖) = 𝑓 (𝑠𝑛−𝑖+1) for all 𝑖 ∈ {1, . . . , 𝑛}.

3. Perform the polynomial fit

˜𝑃𝑛−1 𝑓

through the points (𝑠𝑖, ˜𝑦𝑖) using a Chebyshev expan-

sion (3.17). Note that ˜𝑃𝑛−1 𝑓 will automatically be even. This fit is well-conditioned, and we

neglect the cost of this step.

4. Evaluate the ˜𝑃𝑛−1 𝑓 (0) to be our estimate of ⟨𝑂 (𝑡)⟩.

To summarize, one performs amplitude estimation to acquire the time evolved expectation value at

each Chebyshev node, then performs a polynomial interpolation of the data. The estimate is the

value at 𝑠 = 0.

56

Given an even set of Chebyshev nodes {𝑠1, . . . , 𝑠𝑛}, and making use of Lemma 3.2.1, the

interpolation error 𝐸𝑛−1 assuming perfect data points is given by

|𝐸𝑛−1(0)| ≤

|Tr 𝜌 𝜕𝑛

𝑠 𝑂 𝑠 (𝑡)|

∥𝑂 ∥𝑛!

𝑛
(cid:214)

𝑖=1

|𝑠𝑖 | ≤ max
𝑠∈[−𝑎,𝑎]

∥𝜕𝑛

𝑠 𝑂 𝑠 (𝑡) ∥
∥𝑂 ∥

(cid:16) 𝑎
2𝑛

(cid:17) 𝑛

.

(3.92)

With a suitable bound on 𝜕𝑛

𝑠 𝑂 (𝑡), we can provide an upper bound on the interpolation error at

𝑠 = 0. This bound is provided by the following lemma. In what follows, it will be helpful to define

the parameter

𝑐 := 𝑘 (5/3) 𝑘 𝑚 max
𝑙∈[1,𝑚]

∥𝐻𝑙 ∥𝑡

(3.93)

for ease of notation. This parameter is proportional to the the Hamiltonian size and the "total

Trotter time," meaning the sum of all the forward and backward time steps, in absolute value, for a

2𝑘-th ST formula.

Lemma 3.5.1 (Extrapolation Error Bound for Time-Evolved Observables.). Under the conditions

of Lemma 3.4.1 (𝑐𝑎 ≤ 𝜋/20), the following bounds holds on the Trotterized evolution 𝑂 𝑠 (𝑡) with

step parameter 𝑠 ∈ (0, 𝑎]:

1. for 𝑐 > 𝑛 we have that

∥𝜕𝑛

𝑠 𝑂 𝑠 (𝑡) ∥
∥𝑂 ∥

√︃

(cid:18)

𝑐

<

𝑒3(1 + √︁

8/𝜋𝑒2)

(cid:19) 2𝑛

which gives an interpolation error

|𝐸𝑛−1(0)| <

(cid:18)

129

(cid:19) 𝑛

.

𝑐2𝑎
𝑛

2. For 𝑐 ≤ 𝑛, we have

∥𝜕𝑛

𝑠 𝑂 𝑠 (𝑡) ∥
∥𝑂 ∥

√︂

2𝑛
𝜋

≤

(cid:18) 𝑒4𝑐

(cid:19) 𝑛

2

√

2/𝜋

𝑛!𝑒4𝑐𝑒2

giving an interpolation error

|𝐸𝑛−1(0) ∥ ≤ 2

√

2𝑛 (6𝑐𝑎)𝑛 𝑒24𝑐.

57

The proof of this lemma is a tedious exercise in repeated in the combinatorics of large derivatives

and the triangle inequality, and is left to the end of this chapter. Note that once the derivative bound

holds, the interpolation error bound follows immediately from Lemma 3.2.1.

One motivation for these bounds is deriving asymptotic expressions for the algorithmic com-

plexity. The following theorem gives an asymptotic query complexity for the number 𝑁exp of

Trotter exponentials exp(−𝑖𝐻 𝑗 𝜏).

Theorem 3.5.2. Let 𝑂 (𝑡) = 𝑈†(𝑡)𝑂𝑈 (𝑡) be a time-evolved observable under a Hamiltonian
𝐻 = (cid:205)𝑚
𝑙=1

𝐻𝑙 on 𝑛 qubits, so that 𝑈 (𝑡) = 𝑒−𝑖𝐻𝑡. Suppose there exists a 𝛾 ∈ R+ such that 𝑂/𝛾 can be

block encoded via a unitary 𝑈enc by a state |𝐺⟩ on a set of 𝐿 auxiliary qubits. Let 𝜌 be a quantum

state on 𝑛 qubits, and suppose 𝛾/∥𝑂 ∥ ∈ 𝑂 (1). Then, the number of exponentials 𝑁exp required

to estimate Tr(𝜌𝑂 (𝑡))/∥𝑂 ∥ to precision 𝜖 with confidence 1 − 𝛿 using a 2𝑘 order Suzuki Trotter

formula satisfies

𝑁exp ∈ ˜𝑂

(cid:16)

𝑐 max{𝑐, log(1/𝜖)}𝜖 −1 log(1/𝛿)

(cid:17)

.

Here,

˜𝑂 is big-𝑂 with multiplicative terms suppressed which are logarithmically smaller in 1/𝜖

and 𝑐. Moreover, the number of auxiliary qubits needed is 𝑂 (𝐿).

We give a sketch of the proof. Given a choice of interval [−𝑎, 𝑎] and (even) number of

interpolation points 𝑛, we have from Lemma 3.3.1 that the number of exponentials to perform

evolutions for all Chebyshev nodes goes as

𝑂

(cid:19)

(cid:18) 𝑛 log 𝑛
𝑎

.

(3.94)

However, this is not the total cost since these circuits need to be repeated to perform the appropriate

measurement protocols. Since 𝑂 can be block encoded, the expectation value can be obtained via

an amplitude estimation protocol. By the well-conditioning of Lemma 3.3.3, each data point needs

to be within 𝜖 of the exact Trotter value, up to a logarithmic factor in 𝑛. This robustness is why our

result maintains a (cid:101)𝑂 (𝜖 −1) scaling.

In our proof, we assume the IQAE protocol is used, requiring only a single qubit overhead.

The fractional queries for the noninteger time step also require 𝑂 (1) overhead, meaning the total

58

overhead is 𝑂 (𝐿) due to the block encoding. To relate 𝑛 and 𝑎 to the required precision 𝜖,

simulation time 𝑡 and Hamiltonian 𝐻, Lemma 3.5.1 can be used. Thus, we can relate 𝑁exp to these

basic parameters. We carry out the formal proof in Section 3.8.

As advertised, we see there is a "near-Heisenberg" scaling of 1/𝜖, up to logarithmic factors.

However, there is an unsavory quadratic scaling in the simulation time in cases without high

accuracy demands. I believe this can be improved, because our approach us forces us to have 𝑛

scale linearly in 𝑇, which seems overly pessimistic. Finally, our results suggest the best performance

for using low order formulas, since our bounds are strictly worse for increasing ST order 𝑘.

3.6 Numerical Demonstration

To support the theoretical findings of this chapter, we numerically emulate the polynomial

interpolation procedure on the 1D random Heisenberg model.

I thank James Watson for his

collaboration on these numerics. The Hamiltonian of interest is

𝐻 =

𝑛−1
∑︁

𝑖=0

𝜎𝑖 · 𝜎𝑖+1 + ℎ𝑖 𝑍𝑖

(3.95)

where each ℎ𝑖 ∈ [−ℎ, ℎ] is sampled randomly and uniformly, with ℎ > 0 setting the "disorder

strength." Moreover, 𝜎𝑖 · 𝜎𝑖+1 = 𝑋𝑖 𝑋𝑖+1 + 𝑌𝑖𝑌𝑖+1 + 𝑍𝑖 𝑍𝑖+1 is the dot product of the vector of Paulis

𝜎𝑖 = (𝑋𝑖, 𝑌𝑖, 𝑍𝑖). We take the chain to be a circle, so the 𝑛th qubit is adjacent to the 1st (𝜎𝑛+1 = 𝜎1).

We choose this system because it is simple, yet sufficiently interesting from the perspective of

condensed matter physics, providing a model for the often-studied phenomenon of many-body

localization and closed-system thermalization [25].

Although the product formula simulations are meant to be done on a quantum computer, here

we instead perform all computations classically. Specifically, we compute the product of matrix

exponentials for each product formula. Although not a true quantum simulation, this still provides

accurate information about the Trotter error mitigated by the polynomial interpolation procedure.

To proceed, we first need to decide on a decomposition of the Hamiltonian (3.95), i.e. a partition

of the various terms such that each partition is easy to simulate. To this end, it is helpful to observe

that this system can be represented as a circular graph of 𝑛 nodes, with links representing the

hopping interaction 𝜎𝑖 · 𝜎𝑖+1. The interactions commute if the links don’t meet at a vertex. Thus,

59

we might seek to color the edges of our graph, such that two edges of the same color don’t meet.

If 𝑛 is even, only two colors are necessary for this, by coloring alternate edges. Taking 𝑛 even for

simplicity, we thus partition 𝐻 as

𝐻 = 𝐻even + 𝐻odd + 𝐻pot

(3.96)

where

𝐻even =

∑︁

𝜎𝑖 · 𝜎𝑖+1

𝐻odd =

𝐻pot =

𝑖 even
∑︁

𝜎𝑖 · 𝜎𝑖+1

𝑖 odd
∑︁

𝑖

ℎ𝑖 𝑍𝑖.

(3.97)

Within each partition, the terms commute, and therefore the exponential can be split without error.

As a circuit, each term can be implemented in parallel. Exact methods for computing a two-qubit

unitary can be used [129]. We use the decomposition (3.97) for all of the results in this Section.

Multiple different observables and initial states could be potentially considered. Here we choose

the following:

𝑂 = 𝑍𝑛−1𝑍0𝑋1

|𝜓0⟩ = |1⟩𝑛/2

(3.98)

where |1⟩𝑛/2 is the computational basis state on 𝑛 qubits with a single 1 on qubit 𝑛/2. Intuitively,
we imagine the 1 representing an excitation, and across the circle is the observable of interest 𝑂.

The hopping terms will cause the 1 to "travel" to the other side, but the potential 𝐻pot will cause

more nontrivial behavior. To be concrete, we take the following parameter values in all of the data

presented below: 𝑇 = 1, ℎ = 1, 𝑛 = 6 and 2nd order Trotter.

We first wish to observe the error mitigation in action, as a proof of concept. To do so, need

an "exact" estimate of ⟨𝑂 (𝑡)⟩, meaning an estimate far more accurate than the simulation methods

employed. We use direct matrix exponentiation with 20 digits of precision to accommodate

this need, and all errors are measured with respect to this "exact" calculation. Next, we fix the

60

Figure 3.1 Additive error of the time-evolved expectation value ⟨𝑂 (𝑇)⟩ plotted against maximum
Trotter depth. The seven blue points represent seven distinct numbers of Chebyshev nodes 𝑁
ranging from 2 to 14 in even increments. All data points are calculated to high precision using
matrix multiplication, and thus the dominant error is due to Trotter. The orange denotes the best
single data point gathered for the interpolation, representing the best estimate without classical
post-processing. We see that the polynomial procedure, as the theory suggests, provides an
exponential reduction in Trotter error with each additional data point. Meanwhile, the orange line
appears to, as expected, follow only a polynomial trend.

interpolation interval [−𝑎, 𝑎], with 𝑎 = (∥𝐻 ∥𝑇)−1 a reasonable choice for the largest step size 𝑠.

We then vary the number 𝑁 of interpolation points, and for each 𝑁 (from 2 to 14 in even increments)

we perform the interpolation procedure and calculate the error. By comparing this error with the

error of the "best" data point, i.e., the data point with the smallest Trotter step size, we can get a

sense of how much the Trotter error is mitigated.

Figure 3.1 shows the results of this numerical experiment. Each blue point, going left to right,

represents the Chebyshev interpolation estimate of ⟨𝑂 (𝑡)⟩ with increasing number of interpolation

points 𝑁, while the corresponding orange point is the best single data point. We see that the inter-

polation post-processing dramatically improves the accuracy of the expectation value calculation.

The downward trend continues until flatlining where numerical roundoff takes over as the dominant

error.

One respect in which the results of Figure 3.1 are limited is that data points will not be computed

61

02040608010010−1810−1410−1010−610−2MaximumTrotterDepthTrotterErrorforObservableMitigatedTrotterErrorvs.CircuitDepthExtrapolatedErrorDirectlyMeasuredErrorto near perfection (or be limited by digital round off) even in the case of perfect time evolution.

Expectation values must be estimated through a quantum measurement protocol which can achieve,

at best, a Heisenberg limited scaling of 𝑂 (1/𝜖data) for the number of operations needed to reach

precision 𝜖data. And while Chebyshev interpolation provides robustness to data errors, a numerical

demonstration of this is desirable.

To work within our classical setup, we model data imperfection as Gaussian noise of fixed

width on top of the numerically computed value at each 𝑠. We change this noise parameter 𝜖data

and observe the effect on the true error 𝜖 in the final interpolation estimate. Figure 3.2 gives the

results for various degrees 𝑁 of Chebyshev interpolation. We plot in terms of inverse errors, so

that moving up along either axis corresponds to increased precision in the data or final estimate.

We observe, for each 𝑁, two regimes: a linearly-sloped regime at low data precision and a plateau

for sufficiently precise Trotter data. The first regime corresponds to the error in the final estimate

being dominated by errors in the measurement itself, rather than the systematic Trotter error. In this

regime, the Trotter error has been effectively mitigated, and only the data error remains. However,

for each 𝑁 there is a crossover point where the Trotter error can no longer be brought smaller than

𝜖data. At this point, further increases in data precision no longer improve the final estimate, because

the dominant source is fundamentally the Trotter error. Unsurprisingly, larger 𝑁 increases the

mitigation of Trotter error, delaying this crossover point and improving the final estimate. Beyond

verifying our interpolation approach, this graph illustrates how, for measurement data gathered on

Trotter simulations, there is little value in achieving measurement accuracy beyond the crossover

point. Absent hardware error, Trotter mitigation appears to remove barriers to higher accuracies

through more precise data acquisition.

We haven’t said much about the effects of simulation time, and we conclude with a brief

investigation into this point. Figure 3.3 shows the error in interpolation calculations across four

orders of magnitude for the simulation time. Other fundamental simulation parameters are held

fixed, but following our definitions above, our interpolation interval [−𝑎, 𝑎] will change and hence

the simulation cost. For all values of 𝑇 considered, we see the exponential decay in error leading,

62

Figure 3.2 Error in the expectation value plotted against data noise for several different degrees
𝑁 − 1 of Chebyshev interpolation. Going up along the y axis indicates improved performance.
For small values of 1/𝜖data (large data errors), the final estimate error essentially tracks the data
error, and the Trotter error is negligible. Once 𝜖data is brought under a certain threshold, the final
error flatlines to some plateau, indicating that the final error is dominated by Trotter
(interpolation) error. As 𝑁 increases, the crossover point happens at smaller 𝜖, and the final
estimate is more accurate.

ultimately, to the plateau of machine precision. As 𝑇 increases, the rate remains exponential but

decreases in rate.

3.7 Discussion

In this chapter, we considered the mitigation of Trotter error by use of a standard numerical

technique: polynomial interpolation. This approach is inspired by multiproduct formulas, which

systematically cancel errors due to nonzero Trotter step. Here, however, we are cancelling the

errors "offline", i.e. following the measurements of dynamical observables. This offers accuracy

improvements without enormous quantum overhead, which is especially important as near-term

quantum hardware is noisy and limited. Classical resources, though perhaps less powerful, are

relatively abundant and cheap.

It is interesting to consider to what extent classical resources can chip away at the advantages

of post-Trotter methods over product formulas. The fact that time evolution is only a piece of a full

simulation has allowed us to achieve an exponential reduction in the Trotter error for measuring

63

1001031061091012101510110310510710910111/ϵdata1/ϵTotalErrorScalingvsMeasurementErrorScalingN=2N=4N=6N=8Figure 3.3 Chebyshev approximation error for the observable (3.98), plotted against Chebyshev
degree for various total simulation times 𝑇. Across several orders of magnitude for 𝑇 we observe
an exponential decrease in the algorithmic error with respect to the number of data points until
floating-point precision is reached. However, the decay is slower for larger times.

dynamical observables. A natural follow up question is whether classical techniques, coupled

with first-order Trotter, can reduce the dependent on simulation time 𝑇 to near-linear. An intuitive

argument against this may be that such a scheme would require evolutions much less than 𝑇, and

extrapolating the behavior to later times would be infeasible. This question is likely intimately

related to the 𝑂 (𝑇 2) dependence we’ve derived for our method. Numerical studies may ultimately

shed light as to whether the 𝑇 2 scaling is "real" and whether it can be improved.

Polynomial interpolation is by no means the only approach to understanding the functional

relation 𝑓 (𝑠) between the Trotter step 𝑠 and the quantity of interest. We could also apply Richardson

extrapolation to 𝑓 to estimate 𝑓 (0) given nearby points. Alternatively, rational approximations or

machine learning techniques could be used to uniformly approximate 𝑓 (𝑠). Extending our tests with

the randomized Heisenberg model to include, say, Richardson extrapolation would be beneficial to

facilitate this comparison.

64

5101520253010−1810−1410−1010−610−2DegreeofChebyshevApproximantErrorinFinalApproximationExtrapolatedErrorforDifferentSimulationTimesT=0.1T=1T=10T=1003.8 Proofs

We now provide proofs for Lemma 3.5.1 and Theorem 3.5.2, whose statements were given in

Section 3.5.

Proof of Lemma 3.5.1. For scalar functions 𝑓 (𝑠), derivatives of exp( 𝑓 (𝑠)) can be expressed through

the complete Bell polynomials via Faà di Bruno’s formula.

𝑠 𝑒 𝑓 (𝑠) = 𝑌𝑛 ( 𝑓 ′(𝑠), 𝑓 ”(𝑠), . . . , 𝑓 (𝑛) (𝑠))𝑒 𝑓 (𝑠)
𝜕𝑛

(3.99)

For operator exponentials such as exp(−𝑖 ˜𝐻𝑠𝑡), derivatives can be expressed via repeated application

of Duhamel’s formula. Yet these expressions are always upper bounded by the commuting (scalar)

case [133], so that

∥𝜕𝑛

𝑠 𝑒−𝑖 ˜𝐻𝑠𝑡 ∥ ≤ 𝑌𝑛

(cid:16)

𝑡 ∥𝜕𝑠 ˜𝐻𝑠 ∥, 𝑡 ∥𝜕2

𝑠 ˜𝐻𝑠 ∥, . . . , 𝑡 ∥𝜕𝑛

𝑠 ˜𝐻𝑠 ∥

(cid:17)

.

(3.100)

Note that the exponential disappeared in the bound since it has norm one. Applying Lemma 3.4.1

and invoking the fact that 𝑌𝑛 is monotonic in each argument, this is upper bounded by

(cid:16)

𝑌𝑛

(2 𝑗 𝑗 (𝑒2𝑐) 𝑗+1)𝑛

𝑗=1

(cid:17)

.

An explicit formula for this is given by

(cid:16)

𝑌𝑛

(2 𝑗 𝑗 (𝑒2𝑐) 𝑗+1)𝑛

𝑗=1

(cid:17)

=

∑︁

𝐷

𝑛!
𝑑1! . . . 𝑑𝑛!

𝑛
(cid:214)

𝑗=1

(cid:19) 𝑑 𝑗

(cid:18) 2 𝑗 𝑗 (𝑒2𝑐) 𝑗+1
𝑗!

where 𝐷 is a sum over all indices (𝑑 𝑗 )𝑛

𝑗=1 such that 𝑑 𝑗 ≥ 0 and

Using a Stirling-type bound

𝑛
∑︁

𝑗=1

𝑑 𝑗 𝑗 = 𝑛.

1
𝑗!

≤

(cid:19) 𝑗

(cid:18) 𝑒
𝑗

1
√
2𝜋

65

(3.101)

(3.102)

(3.103)

(3.104)

allows us to write

(cid:16)

𝑌𝑛

(2 𝑗 𝑗 (𝑒2𝑐) 𝑗+1)𝑛

𝑗=1

(cid:17)

≤

∑︁

𝐷

𝑛!
𝑑1! . . . 𝑑𝑛!

(cid:32)√︂

𝑛
(cid:214)

𝑗=1

(cid:33) 𝑑 𝑗

2
𝜋

𝑒 𝑗 (𝑒2𝑐) 𝑗+1

= (𝑒3𝑐)𝑛 ∑︁

𝐷

𝑛!
𝑑1! . . . 𝑑𝑛!

= (𝑒3𝑐)𝑛𝑌𝑛 (√︁

2/𝜋𝑒2𝑐, √︁

(cid:32)√︂

𝑛
(cid:214)

(cid:33) 𝑑 𝑗

𝑒2𝑐

2
𝜋

𝑗=1
2/𝜋𝑒2𝑐, . . . , √︁

2/𝜋𝑒2𝑐)

(3.105)

= (𝑒3𝑐)𝑛𝐵𝑛 (√︁

2/𝜋𝑒2𝑐).

In the second line we brought out 𝑛 factors of 𝑒𝑐 using the condition on the indices 𝐷, and we

identified 𝑌𝑛 evaluated the same at every argument to be the single-variable Bell (or Touchard)
polynomial 𝐵𝑛. We can bound the size of 𝐵𝑛 (√︁2/𝜋𝑒2𝑐) [5] by

𝐵𝑛 (√︁

2/𝜋𝑒2𝑐) ≤

(cid:32)

𝑛
log(1 + √︁ 𝜋
2

𝑛/(𝑒2𝑐))

(cid:33) 𝑛

for all 𝑛 > 0, with 𝑛 = 0 defined by the limit (which is 1). With this,

∥𝜕𝑛

𝑠 𝑒−𝑖 ˜𝐻𝑠𝑡 ∥ ≤

(cid:32)

𝑒3𝑐𝑛
log(1 + √︁ 𝜋
2

𝑛/(𝑒2𝑐))

(cid:33) 𝑛

≤

(cid:18) 𝑒3𝑐𝑛

(cid:19) 𝑛 (cid:32)

2

(cid:33) 𝑛

√︂

8
𝜋

𝑒2𝑐
𝑛

1 +

(3.106)

(3.107)

where we’ve used the bound 1/log(1 + 𝑥) ≤ 1/2 + 1/𝑥. Again, this inequality is valid for 𝑛 = 0 via

the limit, which is always one.

With this bound on the ST formula derivatives, we now turn to bounding 𝜕𝑛

𝑠 𝑂 𝑠 (𝑡). Applying

the binomial theorem and triangle inequality to (3.5),

(cid:19)

∥𝜕 𝑝

𝑠 𝑒𝑖𝑡 ˜𝐻𝑠 ∥ ∥𝜕𝑛−𝑝

𝑠

𝑒−𝑖𝑡 ˜𝐻𝑠 ∥

∥𝜕𝑛

𝑠 𝑂 𝑠 (𝑡)∥
∥𝑂 ∥

≤

≤

𝑛
∑︁

(cid:18)𝑛
𝑝

𝑝=0
(cid:18) 𝑒3𝑐

(cid:19) 𝑛 𝑛
∑︁

𝑝=0

(cid:19)

(cid:18)𝑛
𝑝

𝑝 𝑝 (𝑛 − 𝑝)𝑛−𝑝

(cid:32)

√︂

1 +

8
𝜋

𝑒2𝑐
𝑝

(cid:33) 𝑝 (cid:32)

√︂

1 +

8
𝜋

𝑒2𝑐
𝑛 − 𝑝

(cid:33) 𝑛−𝑝

.

(3.108)

2

At this point, it will be fruitful to consider two regimes. Recall that 𝑐 encodes information about

the simulation time.

66

In the case where 𝑐 > 𝑛, we have

∥𝜕𝑛

𝑠 𝑂 𝑠 ∥
∥𝑂 ∥

(cid:18) 𝑒3𝑐

(cid:19) 𝑛 𝑛
∑︁

2

(cid:18) 𝑒3𝑐

(cid:19) 𝑛

2

𝑝=0
𝑐𝑛 (cid:16)

≤

≤

(cid:19)

(cid:18)𝑛
𝑝

(𝑐 + √︁

8/𝜋𝑒2𝑐) 𝑝 (𝑐 + √︁

8/𝜋𝑒2𝑐)𝑛−𝑝

1 + √︁

8/𝜋𝑒2(cid:17) 𝑛 𝑛

∑︁

𝑝=0

(cid:19)

(cid:18)𝑛
𝑝

√︃

(cid:18)

𝑐

=

𝑒3(1 + √︁

8/𝜋𝑒2)

(cid:19) 2𝑛

.

This implies a relative error in the polynomial fit bounded by

|𝐸𝑛−1(0)| <

(cid:18)

129

(cid:19) 𝑛

.

𝑐2𝑎
𝑛

In the case where 𝑐 ≤ 𝑛, the approximation

(cid:32)

1 + 𝑒2

(cid:33) 𝑝

√︂

8
𝜋

𝑐
𝑝

√

8/𝜋

< 𝑒𝑐𝑒2

holds and is not so crude. Applying this to (3.108),

∥𝜕𝑛

𝑠 𝑂 𝑠 ∥
∥𝑂 ∥

≤

(cid:18) 𝑒3𝑐

(cid:19) 𝑛

2

𝑛!

𝑛
∑︁

𝑝=0

𝑝 𝑝
𝑝!

(𝑛 − 𝑝)𝑛−𝑝
(𝑛 − 𝑝)!

√

2/𝜋.

𝑒4𝑐𝑒2

Regrouping and employing a Stirling bound where appropriate,

∥𝜕𝑛

𝑠 𝑂 𝑠 ∥
∥𝑂 ∥

≤ 𝑒4𝑐𝑒2

√

2/𝜋

(cid:18) 𝑒3𝑐

(cid:19) 𝑛

2

≤ 𝑒4𝑐𝑒2

√

2/𝜋

(cid:18) 𝑒4𝑐

(cid:19) 𝑛

2

2

𝑛! (cid:169)
(cid:173)
(cid:171)
(cid:32)

𝑛!

√

1
√︁𝑝(𝑛 − 𝑝)

(cid:170)
(cid:174)
(cid:172)

√

𝑒𝑛
2𝜋𝑛

+

𝑒𝑛
2𝜋
√

+

2
2𝜋𝑛
(cid:19) 𝑛

𝑛−1
∑︁

𝑝=1
(cid:33)
𝑛 − 1
2𝜋
√

2/𝜋

1
2𝜋
√︂

≤

≤

√

8𝜋 +

(

√

𝑛 − 1)

(cid:18) 𝑒4𝑐

𝑛!𝑒4𝑐𝑒2

2𝑛
𝜋

(cid:18) 𝑒4𝑐

(cid:19) 𝑛

2

𝑛!𝑒4𝑐𝑒2

2
√

2/𝜋.

After another Stirling bound, this gives a corresponding interpolation error of

√

𝐸𝑛−1(0) < 2

2𝑛 (6𝑐𝑎)𝑛 𝑒24𝑐.

67

(3.109)

(3.110)

(3.111)

(3.112)

(3.113)

(3.114)

□

Proof of Theorem 3.5.2. Let 𝑓 (𝑠) = ⟨𝑂 𝑠 (𝑡)⟩/∥𝑂 ∥ be the normalized expectation value under Trot-

ter evolution. Our interpolation algorithm produces an estimate ¯𝑓 of 𝑓 (0) which we require to be

accurate within 𝜖.

| 𝑓 (0) − ¯𝑓 | ≤ 𝜖

(3.115)

There is the interpolation error from the polynomial 𝑃𝑛−1 𝑓 fitting 𝑓 assuming perfect inter-

polation points (𝑠𝑖, 𝑓 (𝑠𝑖)). But 𝑓 (𝑠𝑖) can only be estimated; let’s call ˜𝑦𝑖 this estimate. The error

in ˜𝑦𝑖 in our analysis comes from the statistical error inherent in the estimation protocol as well as

the error in the fractional query procedure for a 1/𝑠 evolution. We can independently consider the

interpolation error and the data error via the triangle inequality.

| 𝑓 (0) − ¯𝑓 | ≤ | 𝑓 (0) − 𝑃𝑛−1 𝑓 (0)| + |𝑃𝑛−1 𝑓 (0) − ˜𝑃𝑛−1 𝑓 (0)|

(3.116)

≤ 𝜖int + 𝐿𝑛𝜖data

Here 𝐿𝑛 is the Lebesgue constant of the interpolation, essentially a condition number, and 𝜖data is

an upper bound on the error in the data.

˜𝑃𝑛−1 𝑓 is the fit to the imperfect data and 𝑃𝑛−1 𝑓 the fit to

the perfect data (𝑠𝑖, 𝑓 (𝑠𝑖)). For generic interpolation nodes, 𝐿𝑛 can grow rapidly; however, for the

set of Chebyshev nodes we obtain a near-optimal value [114].

𝐿𝑛 ≤

2
𝜋 log(𝑛 + 1) + 1

Since we want the total error to be within a threshold 𝜖, we can require

𝜖data ≤

𝜖
2𝐿𝑛

,

𝜖int ≤

.

𝜖

2

(3.117)

(3.118)

Given these error bounds, we can now turn to the cost of acquiring the data points. Because 𝑂/𝛾

can be block encoded, the expectation value calculation can be encoded as an amplitude estimation

problem. Specifically, a Hadamard test circuit gives the amplitude

1 + ⟨𝑂 𝑠𝑖 (𝑡)⟩/𝛾
2

.

(3.119)

If we estimate this amplitude to within accuracy 𝜖data∥𝑂 ∥/2𝛾, we can estimate 𝑓 (𝑠𝑖) within 𝜖data.

Using Iterative Quantum Amplitude Estimation [56], we can obtain this estimate using a Grover

68

iterate 𝐺 constructed from two Hadamard test oracles. The number of Grover oracles 𝑁𝐺 required

is given by

𝑁𝐺 ≤

200𝛾𝐿𝑛
∥𝑂 ∥𝜖data

log

(cid:18) 2𝑛

𝛿 log2

(cid:19)(cid:19)

(cid:18) 𝛾𝐿𝑛𝜋
∥𝑂 ∥𝜖data

(3.120)

to ensure probability 1 − 𝛿 of all data being within 𝜖data of the true value. Each 𝐺 requires two

Hadamard tests, and each Hadamard oracle calls a (controlled) ST evolution once. The number of

controlled exponentials needed for a single data point at value 𝑠𝑖 is in

𝑂

(cid:18) 𝑁𝑘
|𝑠𝑖 |

(cid:19)

log 1/𝜖data

(3.121)

where 𝑁𝑘 = 2𝑚5𝑘−1, and where the logarithm comes from the need for fractional queries with

QSVT. There is also a 𝑂 (1) overhead associated with the fractional queries and IQAE. Altogether,

the number of exponentials for a single data point is in

(cid:18)

𝑂

𝑁𝐺 × 2 ×

log 1/𝜖data

(cid:19)

.

𝑁𝑘
|𝑠𝑖 |

(3.122)

Therefore, the total number 𝑁exp of generating all 𝑛/2 data points (we only need half due to

symmetry) is in

(cid:32)

𝑁exp ∈ 𝑂

𝑁𝐺 𝑁𝑘

(cid:33)

log(1/𝜖data)

.

𝑛/2
∑︁

𝑖=1

1
𝑠𝑖

Plugging in (3.120) for 𝑁𝐺 above and summing over 1/𝑠𝑖 using Lemma 3.3.1,

𝑁exp ∈ 𝑂

(cid:18) 𝛾𝑁𝑘 𝐿𝑛𝑛
∥𝑂 ∥𝜖 𝑎

(log 𝑛) log

(cid:18) 2𝑛

𝛿 log2

(cid:19)(cid:19)

(cid:18) 𝛾𝐿𝑛𝜋
∥𝑂 ∥𝜖

(cid:19)

log(1/𝜖data)

⊂ ˜𝑂

(cid:16) 𝑛
𝑎𝜖 log 1/𝛿

(cid:17)

(3.123)

(3.124)

where ˜𝑂 suppresses factors logarithmic in 𝑛 and 𝜖. We also employed our assumption that

𝛾/∥𝑂 ∥ ∈ 𝑂 (1). The number of nodes 𝑛 and the interpolation interval [−𝑎, 𝑎] will be determined

by 𝜖int, the interpolation error assuming perfect data. To apply our error bounds from the previous

subsection, choose 𝑎 to satisfy Lemma 3.4.1, i.e. 𝑐𝑎 < 𝜋/20, while also taking 1/𝑎 ∈ 𝑂 (𝑐).

Choose 𝑛 ≥ ⌈𝑐⌉. Then the second bound of Lemma 3.5.1 holds. From the interpolation error,

we must satisfy

√
2

2𝑛(6𝑐𝑎)𝑛𝑒24𝑐 < 𝜖/2

(3.125)

69

which in turn can be satisfied provided that

√

2𝑛

4

(cid:16)

6𝑒24𝑐𝑎

(cid:17) 𝑛

< 𝜖

(3.126)

since 𝑛 ≥ 𝑐. Choose 𝑎 such that 6𝑒24𝑐𝑎 = 1/2, which is consistent with our previous conditions

on 𝑎. Then, to satisfy the error bound, 𝑛 can satisfy

√

2𝑛2−𝑛 < 𝜖 .

4

This can be solved using the −1 branch of the LambertW function.

LambertW−1

(cid:16)

−𝜖 log 2/(4

√

2))

(cid:17)

log 2

𝑛 > −

The appropriate asymptotics is 𝑛 ∈ 𝑂 (log(1/𝜖). By taking 𝑛 = 𝑛∗ where

(cid:40)

(cid:38)

𝑛∗ = max

⌈𝑐⌉,

−

LambertW−1(−𝜖 log 2/(4

√

2)

(cid:39)(cid:41)

log 2

∈ 𝑂 (max{𝑐, log(1/𝜖)})

we satisfy all required constraints and arrive at our final asymptotic scaling.

𝑁exp ∈ ˜𝑂

(cid:16)

max{𝑐, log(1/𝜖)}𝑐𝜖 −1 log(1/𝛿)

(cid:17)

(3.127)

(3.128)

(3.129)

(3.130)

(3.131)

□

In the proof above, we set 𝑛 > 𝑐 from the beginning, in order to use the second of the two

bounds from Lemma 3.5.1. Together with 1/𝑎 ∈ 𝑂 (𝑐) this condemns us to a suboptimal 𝑐2 scaling

in the large 𝑐 limit. However, using the first bound instead of the second would not help us, since

the 𝑛𝑐2/𝑎 term in that bound must be order one.

70

CHAPTER 4

TIME DEPENDENT HAMILTONIAN SIMULATION THROUGH DISCRETE CLOCK
CONSTRUCTIONS

The previous chapter was entirely concerned with time independent Hamiltonians. However, this

is only a special (though important) case of general time dependent Hamiltonians that can occur in

systems of interest. Interestingly, a reduction from time dependence to time independence is always

possible, though this has not been properly utilized in the Hamiltonian simulation community.

This chapter concerns itself with a discretized version of the reduction, known as the (𝑡, 𝑡′)

method, that is finite dimensional and therefore amenable to computation. We will consider,

as an application of our formalism, a simulation by qubitization of the encoded time dependent

Hamiltonian. After a brief introduction and motivation for time dependent simulations, we review

the standard (𝑡, 𝑡′) formalism. Then we discretize the clock variable suitably and prove asymptotic

error bounds on the accuracy compared to the time ordered operation. Finally, we describe a

possible simulation by qubitization of the full clock system, before ending with some discussion.

The subsequent Chapter 5 makes use of the clock space construction to argue in favor of the

conjecture that certain multiproduct formulas serve as good approximants to 𝑈. However, in that

chapter, the clock space is only used for a theoretical proof, and plays no role in the simulation.

Here, we take the clock space quite seriously and consider what it would take to fully implement it

using the powerful method of qubitization.

Both chapter and the next are subjects of ongoing research. An early preprint has been

posted [133] which will be updated as these projects reach a close.

4.1

Introduction and Motivation

When a fundamental physical law is expressed as a Hamiltonian, it should generally be expected

to be time independent. This conveys the expected constancy of the theory under consideration.

Such expectations are validated, for example, by experiments which seek to measure time variations

in fundamental physical constants, such as ℏ [127]. To date, no such variations have been measured,

though we must remember there is no theoretical reason, beyond elegance, to disallow them.

71

In any case, there are many practical situations in which a quantum system is naturally modeled

as closed (hence unitary), while still exhibiting time-varying laws. For example, we may imagine

impinging a molecule with electromagnetic pulses in a laboratory setting. These pulses may be

considered large enough to be unaffected by the state of the molecule, yet the molecule is certainly

influenced by the pulse train. Since the amplitude of this pulse may vary with time, so will the

Hamiltonian describing the system. If the state of the quantum system did have an effect on the

incoming electrical pulse, a distinct formalism of open system dynamics should be invoked. Even

if the dynamics aren’t inherently time-varying, it is often useful in the mathematics to shift to an

interaction picture. When a Hamiltonian 𝐻 = 𝐻0 + 𝐻1 has two pieces as shown, where 𝐻0 is

some baseline, "trivial" Hamiltonian that we know how to solve, moving into a frame of reference

"rotating" with respect to 𝐻0 generates a new time dependent 𝐻𝐼 (𝑡) that encapsulates the nontrivial

part. Such as splitting in 𝐻 is common in perturbation theory, where we imagine 𝐻1 small yet

important. Thus, the study of time independent 𝐻 can benefit from understanding how to simulate

time dependent Hamiltonians.

As often occurs in physics, symmetries (in this case related to time translation invariance) lead to

useful simplifications. We have seen in Section 2.5 how for time independent 𝐻, the time evolution

operator 𝑈 (𝑡) = 𝑒−𝑖𝐻𝑡 is a simple matrix exponential, rather than the more nuanced time-ordered

exponential of equation (2.18). Moreover, the existence of stationary states in the time independent

setting provides a useful characterization of the "allowed energies" of a system, and a set of states

whose dynamics are trivial. Despite these simplification, the study of quantum systems with time

independent Hamiltonians is sufficiently rich as to warrant its own focused study.

In Chapter 2, we provided examples of time independent Hamiltonian simulation being used

in generic quantum algorithms applications, particularly linear systems solvers. There are also

applications for time dependent Hamiltonian simulations, such as adiabatic evolution. Generally

speaking, a process is ‘adiabatic’ if it involves the slow transformation of parameters from one value

to another. Here the meaning of "slow" depends on the nature of the problem at hand. Specifying

to Hamiltonians, an adiabatic evolution is a dynamical evolution under a Hamiltonian 𝐻 (𝑡) that

72

changes slowly with time. A standard example is the linear adiabatic evolution

𝐻 (𝑡) =

(cid:16)

1 −

(cid:17)

𝑡
𝑇

𝐻0 +

𝑡
𝑇

𝐻1

(4.1)

which, from 𝑡 = 0 to 𝑡 = 𝑇, changes the Hamiltonian from 𝐻0 to 𝐻1. By "slow" we mean 𝑇 is

much larger than the smallest gap min𝑡 1/𝛿𝐸 (𝑡) between two eigenenergies which overlap the state

of interest. Interestingly, such adiabatic evolutions approximately preserve eigenstates, in the sense

that, starting and initial state as an eigenstate |𝐸0⟩ of 𝐻0, the final state evolved under 𝐻 (𝑡) will be

approximately |𝐸1⟩, and eigenstate of 𝐻1. This is especially useful when trying to prepare ground

states of a nontrivial Hamiltonian 𝐻1 given a simpler one 𝐻0. While the adiabatic approach is

valuable for physical applications [128, 131], it can also enable solutions to optimization problems

when we think of the ground state energy as minimizing, or "optimizing," a function. One can

imagine the ground state of 𝐻1 exhibiting information related to an optimization problem, unrelated

to a physics context. For example, take Quadratic Unconstrained Binary Optimization (QUBO),

which seeks a binary vector 𝑥 ∈ {−1, 1}𝑛 that minimizes

𝑥𝑖𝑄𝑖 𝑗 𝑥 𝑗

∑︁

𝑖 𝑗

(4.2)

for some real symmetric matrix 𝑄. The solution can be encoded as a computational basis state |𝑥⟩

that is the ground state of the Hamiltonian

𝐻𝑄 =

∑︁

𝑖 𝑗

𝑍𝑖𝑄𝑖 𝑗 𝑍 𝑗 .

(4.3)

While this highlights the utility of ground state preparation for general optimization, recent literature

points to other methods, such as Quantum Imaginary Time Evolution, as preferred for these sorts of

tasks as opposed to adiabatic preparation [6, 142]. Nevertheless, it highlights how time dependent

simulations can manifest in broader algorithmic settings.

Because time independent Hamiltonians are special cases of time dependent ones, any algo-

rithms for general 𝐻 (𝑡), and analysis thereof, immediately specializes to an algorithm and analysis

for time independent case. While true, in practice our grasp of time independent simulation algo-

rithms outstrips knowledge of the time dependent case. For example, qubitization is only viable for

73

simulating time independent Hamiltonians, as its basis of operation is a polynomial approximation

to 𝑓 (𝜆) = 𝑒−𝑖𝜆𝑇 on the interval [0, 𝑇] [88]. One could proceed by approximating 𝑈 (𝑇, 0) via

evaluating the expression (2.18) for fixed, large 𝑛. However, the resulting algorithm is, in general,

quite inferior to its time independent version, since the errors from the truncation wash out any

gains in precision. As of writing, there is no time dependent simulation algorithm which saturates

known lower bounds.

Product formulas can be used in a similar manner, applying the formula to each term in the

truncation. This is perhaps the simplest approach to time dependent simulation on a quantum

computer. Because largely fluctuating Hamiltonians will require finer time meshes, the final

cost will depend on the smoothness of the Hamiltonian and the size of derivatives within the mesh

points [137]. More general time dependencies can be handled, without smoothness requirements, by

a certain generalization of the Trotter scheme in which the time integrals (without time ordering) are

retained [105]. The dropping of the smoothness requirement demonstrates that a much broader class

of Hamiltonians is feasibly simulatable. Within their arguments, the authors use a generalization of

product formulas in which the time ordered exponential is split, but the integrals are retained [70].

In this chapter, we will consider an entirely distinct generalization of the product formula in the

time dependent setting. In fact, we will show that it relates to a standard product formula in an

augmented clock space.

4.2 The Clock Space

Mathematically, and more broadly than the Hamiltonian setting, the distinction between time

dependent vs independent systems can be cast as a distinction between autonomous and nonau-

tonomous dynamical systems. Dynamical are differential equations in a single evolution parameter

𝑡, which can always be cast as a first-order initial value problem

(cid:164)𝑥 = 𝑓 (𝑥, 𝑡),

𝑥(0) = 𝑥0

(4.4)

possibly by standard reduction-of-order techniques. Here 𝑥 ∈ R𝑛 consists of 𝑛 evolution parameters

which implicitly depend on time 𝑡. Reasonable smoothness conditions on 𝑓 may be imposed. The

autonomous case corresponds to 𝑓 being independent of 𝑡. Such equations are valuable because they

74

admit a geometric description in terms of phase space and flows. Classically, time independent

Hamiltonian dynamics are a special case of autonomous systems, and the relevant initial value

problem is given by Hamilton’s equations.

( (cid:164)𝑞, (cid:164)𝑝) =

(cid:18) 𝜕𝐻
𝜕 𝑝

, −

(cid:19)

,

𝜕𝐻
𝜕𝑞

(𝑞(0), 𝑝(0)) = (𝑞0, 𝑝0)

(4.5)

This system is autonomous when 𝐻 is independent of the evolution parameter 𝑡.

It has long been recognized that a simple transformation allows for the reduction of nonau-

tonomous systems to autonomous ones [60]. The trick is to promote 𝑡 to a coordinate, thereby

making 𝑓 (𝑥, 𝑡) satisfy the requirement of only depending on coordinates. Letting 𝑠 take the place

of the evolution parameter (time), we still want 𝑡 and 𝑠 to be essentially the same. This is supplied

by the simple equation

(cid:164)𝑡 =

𝑑𝑡
𝑑𝑠

= 1,

𝑡 (0) = 0.

With this, we have the following autonomous system

( (cid:164)𝑥, (cid:164)𝑡) = ( 𝑓 (𝑥, 𝑡), 1),

(𝑥(0), 𝑡 (0)) = (𝑥0, 0)

(4.6)

(4.7)

whose solution encodes the solution to the original (4.4). While it seems that not much has been

gained, this framework finds much application in the consideration of periodically driven systems.

In this case, a cylindrical phase space, with 𝑡 representing the angle, allows for interesting geometric

understanding of the dynamics.

When a classical Hamiltonian 𝐻 (𝑡) is time dependent (nonautonomous), can we perform a

similar trick to reduce to the time independent (autonomous) case? Promoting 𝑡 to a coordinate,

we must formally introduce a conjugate momentum −𝐸, choosing notation suggestive of the time-

energy correspondence. We call the full Hamiltonian 𝐾 (𝑞, 𝑝; 𝑡, 𝐸), to distinguish from the original

Hamiltonian 𝐻 (𝑞, 𝑝, 𝑡), and we wish to determine if a suitable 𝐾 exists. As before, we want

𝑑𝑡/𝑑𝑠 = 1, which because of Hamilton’s equations imply

𝜕𝐻
𝜕 (−𝐸)

= 1.

75

(4.8)

We conclude that 𝐾 = −𝐸 + 𝐹 (𝑞, 𝑝, 𝑡) for some 𝐹. But we need 𝐾 to reproduce the same dynamics

for 𝑞, 𝑝 as 𝐻. This implies 𝐹 = 𝐻. Interestingly, the final Hamilton equation

𝑑𝐸
𝑑𝑠

=

𝜕𝐻
𝜕𝑡

(4.9)

corresponds with the expected equation for energy change, up to a sign. This explains our choice

of a minus sign in the conjugate momentum −𝐸.

In summary, we have a prescription for converting nonautonomous Hamiltonians into au-

tonomous ones depending only on coordinates. In hindsight, the way this is accomplished is rather

silly. The coordinate 𝑡 has dynamics completely independent of the values of any other coordinate

or momentum, including 𝐸, and gets pulled in a straight line at a constant velocity. Time marches

forward. As 𝑡 changes, so does 𝐻, which affects all of the other coordinates in the desired way.

This is much like a reel of movie tape moving through the machine at a constant rate to change

what’s on the screen, mimicking the forward flow of time.

What about quantum Hamiltonians? We could try to quantize the above, imposing the usual

commutation relation

[ˆ𝑡, − ˆ𝐸] = 𝑖𝐼.

(4.10)

Absent periodicity, a natural choice of Hilbert space of 𝑡 is 𝐿2( [0, 𝑇]): square integrable functions

on the interval of simulation. We then get a representation of 𝑡 and 𝐸 as 𝑡 multiplication and

𝑡-derivatives, respectively.

(ˆ𝑡𝜓)(𝑡) = 𝑡𝜓(𝑡),

( ˆ𝐸𝜓) (𝑡) = 𝑖𝜕𝑡𝜓

(4.11)

The augmented "clock" Hamiltonian, so named because of the time coordinate 𝑡, has the form

𝐻𝑐 = 𝐻 − 𝑖𝜕𝑡

(4.12)

which looks eerily similar to a rearranged Schrödinger operator. Indeed, if 𝜓sol(𝑡, 𝑞) is a solution

to the Schrödinger equation encoded on the full "clock" Hilbert space, then we see that the state

𝜓𝛼 (𝑡, 𝑞) = 𝑒−𝑖𝛼𝑡𝜓sol,

𝛼 ∈ R

(4.13)

76

is formally an eigenstate of 𝐻𝑐 with eigenvalue 𝛼. As a caution, this state is not properly normalized,

nor is it normalizable unless the simulation interval [0, 𝑇] is finite.

The above manipulations are purely formal, helpful only for calculational or interpretational

purposes. From a physical point of view, 𝐻𝑐 is not bounded from below, allowing for potentially

infinite energy extraction if such as system existed and could be coupled with. In particular, this

system transitions to lower energies as wavepackets travel faster and faster to the left of the clock

space, a seemingly senseless possibility.

The quantum mechanical version of this trick, sometimes called the (𝑡, 𝑡′)-formalism because of

the two distinct "times", finds use in periodically-driven quantum systems [21]. But our purposes,

the elimination of explicit dependence on the evolution parameter is most exciting, because it

implies the time evolution operator requires no time-ordering, while still encoding the full time

dynamics [103]. Thus, while time independent 𝐻 is a special case of time dependence, time

dependent simulation 𝐻 reduces, formally, to time independent simulation on an augmented space.

To understand the nature of the encoding better, we have to be somewhat more careful. Let’s

provide a more concrete description of the situation. The full Hilbert space is given by

H = H𝑠 ⊗ H𝑐

(4.14)

where H𝑐 (cid:27) 𝐿2(M), and M is the (connected) one-dimensional smooth manifold representing 𝑡.

We have considered M (cid:27) [0, 𝑇] here, but we might also consider a circle (periodic dynamics) or

the real line, where translations are a bit more natural. On H𝑐, 𝐸 acts as a generator of translations,

but is an unbounded operator. Nevertheless, the exponentials of 𝐸 above are well defined through

the spectral theorem and functional calculus for unbounded operators [61].

States 𝜓 ∈ H can then be expressed as certain functions on M whose value 𝜓(𝑡) is a state on

H𝑠. The inner product on H is the natural one.

⟨𝜙|𝜓⟩ :=

∫

M

⟨𝜙(𝑡)|𝜓(𝑡)⟩𝑠 𝑑𝑡

(4.15)

Here, ⟨·|·⟩𝑠 denotes the inner product on H𝑠. From now on, we use 𝜏 to denote the evolution

parameter in order to avoid confusion with the clock coordinate 𝑡.

77

The interval 𝐼 over which the dynamics of the quantum system take place needs to be embedded

within M. If 𝐼 is not exactly M, then the definition a time dependent observable 𝐴(𝑡) will need to

be extended to cover the entire clock space. Once done, 𝐴(𝑡) is promoted to an (time independent)

observable on H , denoted A, by acting on 𝜓 ∈ H in a manner corresponding with the original

space.

(A𝜓) (𝑡) := 𝐴(𝑡)𝜓(𝑡)

(4.16)

All such operators A are seen to be local in H𝑐, since they act with simple multiplication.

Having laid the above groundwork, we can return to the question of dynamics. Let H be the

promoted Hamiltonian operator as discussed in the previous paragraph. Let U(𝜏) be the unitary

operator given by

U(𝜏) = 𝑒𝑖𝐸𝜏𝑒−𝑖(H−𝐸)𝜏.

One can verify that U solves the following Schrödinger equation.

Here,

𝑖𝜕𝜏U(𝜏) = H(𝜏)U(𝜏)

U(0) = 𝐼

H(𝜏) ≡ 𝑒𝑖𝐸𝜏H𝑒−𝑖𝐸𝜏

(4.17)

(4.18)

(4.19)

is a 𝜏-dependent Hamiltonian corresponding to simple, uniform translation along the clock space.

For any state Ψ0 ∈ H , the function

Ψ(𝜏) := U(𝜏)Ψ0

(4.20)

solves the Schrödinger equation generated by H(𝜏), but more importantly, it encodes solutions to

the dynamics under 𝐻 (𝑡). Indeed, for any 𝑡 ∈ M, we have a state 𝜓(𝜏, 𝑡) ∈ H𝑠 defined by

𝜓(𝜏, 𝑡) := [Ψ(𝜏)] (𝑡) = [U(𝜏)Ψ0] (𝑡)

(4.21)

78

which solves the Schrödinger equation of interest.

𝑖𝜕𝜏𝜓(𝜏, 𝑡) = 𝑖𝜕𝜏 [U(𝜏)Ψ0] (𝑡)

= [H(𝜏)Ψ0] (𝑡)

= 𝐻 (𝜏 + 𝑡)𝜓(𝜏, 𝑡)

(4.22)

The interpretation is that each 𝑡 constitutes an initial time for performing the simulation, so we have

a family of solutions parametrized by 𝑡 with initial state 𝜓(0, 𝑡). The evolution parameter acts, as

expected, as the total time elapsed in the simulation.

Finally, we can obtain a collection of induced time-evolution operators, 𝑈 (𝑡′ + Δ𝑡, 𝑡′) on H𝑠 for

each 𝑡′ ∈ M. It acts on states 𝜓0 ∈ H𝑠 as follows

𝑈 (𝑡 + 𝜏, 𝑡)𝜓0 = [U(𝜏)Ψ0] (𝑡)

(4.23)

where Ψ0 ∈ H is any state for which Ψ0(𝑡) = 𝜓0. This operator is unitary and solves the Schrödinger

equation in the usual sense. Therefore it is exactly equivalent to the time-ordered exponential of

expression (2.17).

We’ve shown now that the clock space encodes a collection of solutions to the dynamics under

𝐻 (𝜏), one for each initial time 𝑡 with initial state Ψ0(𝑡) ∈ H𝑠. However, one might only care about

one solution, say, at 𝑡 = 0, and the ability to extract that solution from the encoding. Suppose the

desired initial state is |𝜓0⟩ ∈ H𝑠, and our initial time is 𝑡0 ∈ M. The idea is to prepare an initial

product state Ψ0 = 𝜓0 ⊗ 𝜙0 ∈ H , where 𝜙0 ∈ H𝑐 has overlap 1 − 𝛿 in a 𝜖-neighborhood of 𝑡0.

After performing the evolution under U for the desired length 𝜏, perform a measurement of 𝑡. The

probability of measuring 𝑡 within 𝜖 of 𝑡0 + 𝜏 is 1 − 𝛿. Moreover, provided that the variation of 𝐻 (𝑡)

around 𝑡 = 𝑡0 is small for variations of 𝑡 ± 𝜖, any value within the 𝜖-neighborhood will suffice. Of

course, any real measurement of 𝑡 will have an uncertainty width, and this must be brought within

the size of the variation of 𝐻. The actual state will be slightly mixed, but very close to pure.

Preparing the initial state 𝜓0 ⊗ 𝜙0 is just as hard as preparing each separately, so we focus on 𝜙0.

One way to prepare a sharp peaked state is an initial 𝑡 measurement of accuracy 𝜖meas according

to the requirements above. If 𝐻 (𝑡) can be shifted appropriately so that the measurement result

79

aligns with the desired start time, or the state can be shifted, then we’ve successfully prepared the

desired state, and the clock Hamiltonian H − 𝐸 can be turned on. However, no serious attempt

towards an implementation on physical quantum hardware has been made.

In terms of digital

quantum computing, a proposal for simulating the clock system will be supplied in Section 4.4,

which follows our construction of a discrete clock space.

To summarize, we have shown how the propagator 𝑈 generated by a time dependent 𝐻 can be cast

as an ordinary operator exponential, via the inclusion of a 1-dimensional clock space. Interesting in

its own right, this framework also allows for a natural unification of ideas regarding "Trotterization."

This term is used to refer to both (a) the splitting up of an (ordinary) operator exponential of 𝐻 =

(cid:205) 𝑗 𝐻 𝑗 into exponentials of the various 𝐻 𝑗 , or (b) the simulation of a time dependent Hamiltonian

by time independent simulations over small time intervals, indicated in (2.18). These can, in fact

be viewed as manifestations of the same phenomenon: a splitting of operator exponentials. To

illustrate this with an example relevant to this paper, let’s consider a simple symmetric Trotterization

of equation (4.17).

𝑈2(𝑡0 + Δ𝑡, 𝑡0) ≡ 𝑒𝑖𝐸Δ𝑡 (cid:16)

𝑒−𝑖𝐸Δ𝑡/2𝑒−𝑖𝐻 (𝑡0)Δ𝑡𝑒−𝑖𝐸Δ𝑡/2(cid:17)

= 𝑒−𝑖𝐻 (𝑡0+Δ𝑡/2)Δ𝑡

(4.24)

We have just derived the midpoint formula [137, 124] from scratch. The Trotter product theorem

says that

𝑒𝑖𝐸Δ𝑡 (cid:16)

𝑒−𝑖𝐸Δ𝑡/2𝑘 𝑒−𝑖𝐻 (𝑡0)Δ𝑡/𝑘 𝑒−𝑖𝐸Δ𝑡/2𝑘 (cid:17) 𝑘

lim
𝑘→∞

= 𝑈 (𝑡0 + Δ𝑡, 𝑡0)

(4.25)

even though 𝐸 is unbounded [97]. Thus, we can expect that

𝑈 (𝑘)
2

(𝑡0 + Δ𝑡, 𝑡0) ≡ 𝑒𝑖𝐸Δ𝑡 (cid:16)

𝑒−𝑖𝐸Δ𝑡/2𝑘 𝑒−𝑖𝐻 (𝑡0)Δ𝑡/𝑘 𝑒−𝑖𝐸Δ𝑡/2𝑘 (cid:17) 𝑘

(4.26)

constitutes a good approximation to 𝑈 for sufficiently large 𝑘 ∈ Z+ and small Δ𝑡.

This opens up the possibility of a more unified approach to Hamiltonian simulation algorithms,

which has not yet been properly considered. For example, a natural generalization of product

formulas to time dependent 𝐻 could be a regular product formula of 𝐻𝑐 on the enlarged space. For

simplicity, let’s take 𝐻 (𝑡) in whole as a single term, so that 𝐻𝑐 = 𝐻 − 𝐸 has two terms. We can

80

define time dependent product formulas by taking product formulas along these two terms. This

allows us to define, in particular time dependent generalizations of the recursive Suzuki-Trotter

formulas of (2.28). Mathematical difficulties emerge in classifying the approximation order of

these formulas, arising from the unboundedness of 𝐸. We seek to ameliorate this in our current

work, focusing our attention on the 2nd order symmetric formula.

4.3 Finite Clock Spaces

There are several reasons we are motivated to consider a discretization of the clock space

introduced in the previous section. First, any real computation performed using the clock space

will require a finite number of states. A natural choice for discretization is along the time ("position")

basis, and we will consider that here.

There are also formal reasons to consider a finite clock space. We’ve already seen how the

clock space aids in the understanding of time dependent generalization of product formulas. Taking

this idea further, we might consider time dependent multiproduct formulas, i.e. how to construct

MPFs for time dependent problems. To ensure these MPFs work, we would like to show that an

error series exists whose terms can be cancelled order by order through linear combinations of

time dependent product formulas. However, such error series are more difficult to show on generic

separable Hilbert spaces, and moreover the operator 𝐸 is unbounded. By making the clock space

finite, performing the relevant analysis, then taking the limit, we can potentially avoid these. This

programme is described in more detail in Chapter 5, but a full proof of this is an ongoing research

project.

Without further ado, we introduce our finite dimensional clock space, which we will sometimes

call the "clock register." We discretize the clock variable 𝑡 into 𝑁𝑐 = 𝑁 𝑝 × 𝑁𝑞 basis states,

where 𝑁 𝑝 ∈ Z+ will represent the number of Trotter steps used in the simulation. Each "Trotter

step" is further divided into 𝑁𝑞 ∈ Z+ steps for reasons that will be discussed shortly. We label

these orthonormal basis states | 𝑗⟩ for 𝑗 ∈ [0, 𝑁𝑐 − 1] ∩ Z. We will find it useful to consider,

for our purposes, only periodic Hamiltonians. This is natural to understand, since translation

operators like 𝐸 act most naturally on R or the circle (periodic boundary conditions), and the circle

81

is bounded. Nonperiodic Hamiltonians can be accommodated by a simple reflection, defining

𝐻 (𝑇 + 𝑡) := 𝐻 (𝑇 − 𝑡) for 𝑡 ∈ [0, 𝑇]. In our work below, we will want 𝐻 (𝑡) to be a differentiable

bounded function within the grid points, and although the reflection introduces nonsmoothness, we

can simply take one of the grid points to be the midpoint of simulation.

For simplicity, and for lack of a reason otherwise, we will take these grid points (𝑡 𝑗 ) 𝑁𝑐−1
𝑗=0

to

be uniformly spaced over the interval [0, 𝑇]: 𝑡 𝑗 = 𝑇 𝑗/𝑁𝑐 (taking 𝑁𝑐 to be an even integer, so that

the midpoint requirement discussed directly above is satisfied). We let 𝛿𝑡 := 𝑇/𝑁𝑐 denote the grid

width. We also take the natural discretization of 𝐻 (𝑡) onto the clock space

𝐻 (𝑡) ↦→

𝑁𝑐−1
∑︁

𝑗=0

𝐻 𝑗 ⊗ | 𝑗⟩⟨ 𝑗 | ≡ 𝐶 (𝐻)

(4.27)

where 𝐻 𝑗 ≡ 𝐻 (𝑡 𝑗 ). Observe that 𝐶 (𝐻) has no dependence on the evolution parameter, i.e., it is

autonomous. The notation 𝐶 (𝐻) is used to suggest a controlled operation, where the control is on

the clock register.

Choosing the appropriate discretization of 𝐸 is somewhat more tricky, though the choice appears

obvious in hindsight. Since 𝐸 acts as a derivative, it makes sense to take the discretized version to

be a finite difference operator. For example,

Δ := −𝑖

𝑈+ − 𝑈−
2𝛿𝑡

(4.28)

where 𝑈+ is the shift operator defined by 𝑈+| 𝑗⟩ = | 𝑗 + 1⟩ and 𝑈− = 𝑈†

+ is the backwards shift (all

increments taken mod 𝑁𝑐). This is the approach we ultimately take. However, we note that author

of this dissertation, and collaborators, began by considering a distinct approach via the logarithm

of the translation operator

˜Δ = 𝑖 log 𝑈+.

(4.29)

While apparently sensible, given the analogous relation between 𝐸 and shifts on the clock space,

this operator is not nicely behaved. For example, its commutator with the "position operator"

(cid:205) 𝑗 𝑡 𝑗 | 𝑗⟩⟨ 𝑗 |, rather than being near-identity, has long off-diagonal tails. This behavior may be of

independent interest, but from now on we will concern ourselves with Δ as the discrete version of

𝐸.

82

With these choices, our full clock Hamiltonian becomes

𝐻𝑐 := 𝐶 (𝐻) − Δ.

(4.30)

Already, we can show some reasonable properties carry over to this setting.

Lemma 4.3.1. In the notation above, let 𝐻 : [0, 𝑇] → Herm(H ) be a time dependent Hamiltonian

on a finite-dimensional vector space H . Then

[Δ, 𝐶 (𝐻)] = 𝑖 Re

(cid:32)
𝑈+

∑︁

𝑗

𝐻 𝑗+1 − 𝐻 𝑗
𝛿𝑡

(cid:33)

⊗ | 𝑗⟩⟨ 𝑗 |

where Re( 𝐴) := ( 𝐴 + 𝐴†)/2 denotes the Hermitian part of 𝐴.

If 𝐻 is differentiable in each

subinterval with bounded derivative, then we further have

∥ [Δ, 𝐶 (𝐻)] ∥ ≤ max
𝑡∈[0,𝑇]

∥ (cid:164)𝐻 (𝑡) ∥.

We remark here the connections to the canonical commutation relation [ 𝑓 (𝑥), 𝑝] = 𝑖 𝑓 ′(𝑥). The

additional shift by 𝑈+ is a relatively small deviation from a finite difference approximation being

performed on the Hamiltonian.

Proof. We proceed in several steps, first by computing [𝑈+, 𝐶 (𝐻)]. We have

[𝑈+, 𝐶 (𝐻)] =

=

𝑁𝑐−1
∑︁

𝑗=0
𝑁𝑐−1
∑︁

𝑗=0

𝐻 𝑗 ⊗ [𝑈+, | 𝑗⟩⟨ 𝑗 |]

𝐻 𝑗 ⊗ (| 𝑗 + 1⟩⟨ 𝑗 | − | 𝑗⟩⟨ 𝑗 − 1|.

(4.31)

By splitting the sum and reindexing (all increments modulo 𝑁𝑐), we can move the difference to the

𝐻 𝑗 , giving

[𝑈+, 𝐶 (𝐻)] =

∑︁

𝑗

(𝐻 𝑗 − 𝐻 𝑗+1) ⊗ | 𝑗 + 1⟩⟨ 𝑗 |

= −𝑈+

∑︁

𝑗

(𝐻 𝑗+1 − 𝐻 𝑗 ) ⊗ | 𝑗⟩⟨ 𝑗 |.

(4.32)

83

Next, we have that [𝑈−, 𝐶 (𝐻)] = −[𝑈+, 𝐶 (𝐻)]†. Thus,

[𝑈+ − 𝑈−, 𝐶 (𝐻)] = −2 Re

(cid:32)
𝑈+

∑︁

𝑗

(𝐻 𝑗+1 − 𝐻 𝑗 ) ⊗ | 𝑗⟩⟨ 𝑗 |

(4.33)

(cid:33)

and the full result follows almost immediately from the definition of Δ given in equation (4.28).

As for the upper bound, we note that ∥ Re( 𝐴) ∥ ≤ ∥ 𝐴∥ for any finite-dimensional 𝐴, and by

unitary invariance of the spectral norm we have

∥ [Δ, 𝐶 (𝐻)] ∥ ≤

(cid:13)
(cid:13)
∑︁
(cid:13)
(cid:13)
(cid:13)

𝑗

𝐻 𝑗+1 − 𝐻 𝑗
𝛿𝑡

⊗ | 𝑗⟩⟨ 𝑗 |

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

= max
𝑗

(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝐻 𝑗+1 − 𝐻 𝑗
𝛿𝑡

(cid:13)
(cid:13)
(cid:13)
(cid:13)

.

The upper bound then follows from the claim

(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝐻 𝑗+1 − 𝐻 𝑗
𝛿𝑡

(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤ max

𝑡∈[𝑡 𝑗 ,𝑡 𝑗+1]

∥ (cid:164)𝐻 (𝑡) ∥

coming from a the fundamental theorem of calculus and the triangle inequality.

(4.34)

(4.35)

□

Having defined the clock space and Hamiltonian, we wish to prepare a suitable initial state. A

seemingly adequate and natural choice is to take |𝜓0⟩ ⊗ |0⟩, where |𝜓0⟩ is the initial state of the

system of interest and |0⟩ is the clock state at the initial time 𝑡 = 0. However, problems immediately

arise which can be traced to the fact that the continuous version of |0⟩ is 𝛿(𝑡), which is not a

normalizable state vector. This formal problem finds its way into the discrete setting, in that the

finite difference Δ does not properly compute a derivative of |0⟩. Thus, Δ fails to translate |0⟩

properly into later times, and the time dependent simulation fails.

To fix this issue, we take a cue from the continuous setting, where the best we can do is take a

wavepacket of small enough width to suit our purposes. For simplicity, this wavepacket may as well

be Gaussian, with some width 𝜎 to be chosen with care. Thus, we introduce Gaussian functions

𝜙𝜇 (𝑡; 𝜎) =

𝑒−|𝑡−𝜇|2

𝑐/𝜎2

1
√
N

of width 𝜎 ∈ R+ and center 𝜇 ∈ [0, 𝑇). Here |·|𝑐 is the shortest distance to 0 modulo 𝑇,

|𝑡|𝑐 = min {|𝑡|, |𝑇 − 𝑡|}

84

(4.36)

(4.37)

so that, with 0 and 𝑇 identified, 𝜙𝜇 is smooth everywhere except 𝜇 + 𝑇/2 mod 𝑇. Moreover,

N ∈ R+ is chosen such that the discretized vector

|𝜙𝜇⟩ =

∑︁

𝑗

𝜙𝜇 (𝑡 𝑗 ; 𝜎)| 𝑗⟩

(4.38)

is normalized in the Euclidean sense (i.e., a quantum state vector). Technically, N has some

dependence on 𝜇, but in our case we will only consider 𝜇 = 𝑡 𝑗 for some 𝑗, in which case N only

depends on parameters such as 𝑁𝑐 and 𝜎. Because of this choice, we will more simply write

|𝜙 𝑗 ⟩ ≡ |𝜙𝑡 𝑗 ⟩.

We are now ready to more clearly elucidate the overall strategy of the clock space construction.

Figure 4.1 gives a schematic of the relevant components. We imagine 𝑁 𝑝 chunks of time steps,

each containing 𝑁𝑞 subdivisions. As stated above, each of the 𝑁 𝑝 should be thought of as a single

Trotter step in the evolution under 𝐻 (𝑡). The 𝑁𝑞 substates ensure that 𝛿𝑡 is sufficiently small such

that the approximation of Δ to a derivative of 𝜙 𝑗 holds. In particular, we will desire 𝜎 ≫ 𝛿𝑡. On

the other hand, we want the variation of 𝐻 within the envelope of 𝜙 𝑗 to be small. That is, we want

𝜎 < 𝑇/𝑁 𝑝. Because, presumably, we’ve chosen each Trotter step sufficiently small, this ensures

that 𝐻 is approximately constant over the bulk of |𝜙 𝑗 ⟩. Of course, we will want to ensure all of the

above conditions are met using as few resources, such as clock register states, as possible to get an

accurate simulation.

Let’s now characterize the accuracy of this construction. First, it will be helpful to have a

characterization of the size of the normalization N .

Lemma 4.3.2. In the notation above, the normalization constant N ∈ R+ for Gaussian states |𝜙 𝑗 ⟩

peaked at 𝜇 = 𝑡 𝑗 satisfies

∈ 𝑂 (√︁𝛿𝑡/𝜎).

1
√
N

Proof. By cyclicity, the normalization N is the same for all |𝜙 𝑗 ⟩, so we consider 𝑗 = 0. Because

85

Figure 4.1 Schematic of the discrete clock Hilbert space. The clock register has an initially
prepared Gaussian state which is translated uniformly under the clock Hamiltonian. Its location
controls the Hamiltonian applied to the system of interest. The Hamiltonian varies little over each
of the 𝑁 𝑝 large steps, and the Gaussian is wide compared to the 𝑁𝑞 subdivisions within each large
step.

|𝜙0⟩ is normalized in the Euclidean norm, we have

𝑁𝑐−1
∑︁

N =

𝑒−2|𝛿𝑡2 𝑗 |2

𝑐/𝜎2

𝑗=0
𝑁𝑐/2−1
∑︁

𝑗=0

=

𝑒−2 𝑗 2𝛿𝑡2/𝜎2 +

𝑁𝑐−1
∑︁

𝑗=𝑁𝑐/2

𝑒−2(𝑁𝑐− 𝑗)2𝛿𝑡2/𝜎2

= 1 +

𝑁𝑐/2−1
∑︁

𝑗=1

𝑒−2 𝑗 2𝛿𝑡2/𝜎2 +

𝑁𝑐/2
∑︁

𝑗=1

𝑒−2 𝑗 2𝛿𝑡2/𝜎2

=

𝑁𝑐
2 −1
∑︁

𝑗=0

𝑒−2 𝑗 2𝛿𝑡2/𝜎2 +

𝑁𝑐
2∑︁

𝑗=0

𝑒−2 𝑗 2𝛿𝑡2/𝜎2 − 1.

(4.39)

We may lower bound the sums as Riemann approximations to a Gaussian integral, giving error

function erf

N ≥

√︂ 𝜋
8

(cid:18)

erf

(cid:19)

(cid:18)𝑇 + 2𝛿𝑡
√
2𝜎

+ erf

(cid:19)(cid:19)

(cid:18) 𝑇
√

2𝜎

− 1 >

√︂ 𝜋
2

𝜎
𝛿𝑡 erf

(cid:19)

(cid:18) 𝑇
√

2𝜎

− 1 ,

(4.40)

which then implies

≤ √︁

2/𝜋(𝛿𝑡/𝜎)

1
N

1
(cid:17)

√︃ 2
𝜋

𝛿𝑡
𝜎

−

erf

(cid:16) 𝑇
√
2𝜎

√︂

2
𝜋

=

(𝛿𝑡/𝜎) + 𝑂

(cid:18)

(𝛿𝑡/𝜎)

(cid:18) 𝛿𝑡
𝜎

(cid:19)(cid:19)

+ 𝑒− 𝑇2

2𝜎2

The result follows simply from taking a square root.

∈ 𝑂 (𝛿𝑡/𝜎).

(4.41)

□

86

With this technical lemma in hand, we turn to showing that Δ indeed acts as a generator of

translations on the clock space for |𝜙 𝑗 ⟩, provided 𝜎 is large relative to 𝛿𝑡 and that the Gaussian is

not truncated by small 𝑇.

Lemma 4.3.3. In the notation introduced in this section, for any 𝑚 ∈ Z+ we have

𝑒𝑖Δ𝑚𝛿𝑡 |𝜙 𝑗 ⟩ = |𝜙 𝑗+𝑚⟩ + 𝑂

(cid:16)

𝑚(𝛿𝑡/𝜎)2 + 𝑚√︁𝛿𝑡/𝜎𝑒−(𝑇/2𝜎)2 (cid:17)

where the asymptotics 𝑂 are understood to be taken as 𝛿𝑡/𝜎 → 0 and 𝜎/𝑇 → 0.

Proof. Performing a 1st order Taylor expansion of the exponential,

𝑒𝑖Δ𝛿𝑡 |𝜙 𝑗 ⟩ = |𝜙 𝑗 ⟩ + 𝑖𝛿𝑡Δ|𝜙 𝑗 ⟩ + 𝑅1(𝛿𝑡)|𝜙 𝑗 ⟩,

where 𝑅1 is the Taylor remainder operator

𝑅1(𝛿𝑡) = 𝛿𝑡

∫ 𝛿𝑡

0

𝜕2
𝜕𝜏2

𝑒𝑖Δ𝜏𝛿𝑡𝑑𝜏 = −

∫ 𝛿𝑡

0

𝑒𝑖Δ𝜏𝑑𝜏(𝛿𝑡Δ2).

The Taylor error can be bounded, via the triangle inequality for integrals, as

∥𝑅1(𝛿𝑡)|𝜙 𝑗 ⟩∥ ≤ 𝛿𝑡2∥Δ2|𝜙 𝑗 ⟩∥.

The action of Δ on discretized functions |𝑔⟩ of the clock space is given by

(4.42)

(4.43)

(4.44)

Δ |𝑔⟩ = −𝑖

𝑁𝑐−1
∑︁

𝑔(𝑡 𝑗 )

(cid:18) | 𝑗 + 1⟩ − | 𝑗 − 1⟩
2𝛿𝑡

(cid:19)

𝑗=0
𝑔(𝑡 𝑗+1) − 𝑔(𝑡 𝑗−1)
= 𝑖 ∑︁
2𝛿𝑡

𝑗

| 𝑗⟩

(4.45)

= 𝑖 |𝐷𝛿𝑡𝑔⟩ .

Here 𝐷𝛿𝑡 𝑓 (𝑥) :=

𝑓 (𝑥+𝛿𝑡)− 𝑓 (𝑥−𝛿𝑡)
2𝛿𝑡

is the symmetric finite difference of halfwidth 𝛿𝑡 at point 𝑥. Thus,

Δ2|𝜙 𝑗 ⟩ = −|𝐷2

𝛿𝑡 𝜙 𝑗 ⟩. We consider the error of this finite difference in terms of an approximation to
the derivative for values of 𝑡 within 𝑇/2 − 2𝛿𝑡 of 𝑡 𝑗 in circle distance. On this part of the domain,

𝜙 𝑗 (𝑡 ± 2𝛿𝑡) is smooth, hence

|𝐷2

𝛿𝑡 𝜙 𝑗 ⟩ = |𝜕2

𝑡 𝜙 𝑗 ⟩ + 𝑂 (𝛿𝑡2𝜙(4)
𝑗

)

(4.46)

87

where the superscript (4) indicates a fourth derivative. Near the edge of the Gaussian, the second-

derivative property does not hold; however, these parts of the state vector have amplitude which is on
the order 𝑂 (N −1/2𝑒−(𝑇/2𝜎)2), which by Lemma 4.3.2 is 𝑂 (√︁𝛿𝑡/𝜎𝑒−(𝑇/2𝜎)2). This gets multiplied

by 𝛿𝑡−2 due to the second finite difference 𝐷𝛿𝑡 being taken. Taking the two sources independently

as an upper bound, we have

∥Δ2|𝜙 𝑗 ⟩∥ ∈ 𝑂

(cid:16)

𝛿𝑡2/𝜎4 + (𝜎𝛿𝑡3)−1/2𝑒−(𝑇/2𝜎)2 (cid:17)

(4.47)

where 𝜎−4 comes from the four derivatives of the Gaussians. Thus, the total Taylor remainder may

be upper bounded using (4.43) as

∥𝑅1(𝛿𝑡)|𝜙 𝑗 ⟩∥ ∈ 𝑂

(cid:16)

(𝛿𝑡/𝜎)4 + √︁𝛿𝑡/𝜎𝑒−(𝑇/2𝜎)2 (cid:17)

.

(4.48)

To complete the proof we return to the linear Taylor expansion in (4.42). Using similar reasoning

to above,

|𝜙 𝑗 ⟩ + 𝑖𝛿𝑡Δ|𝜙 𝑗 ⟩ = |𝜙 𝑗 ⟩ − 𝛿𝑡|𝐷𝛿𝑡 𝜙 𝑗 ⟩

= |𝜙 𝑗 ⟩ − 𝛿𝑡|𝜕𝑡 𝜙 𝑗 ⟩ + 𝑂

(cid:16)

(𝛿𝑡/𝜎)2 + √︁𝛿𝑡/𝜎𝑒−(𝑇/2𝜎)2 (cid:17)

.

(4.49)

Finally, what remains is a linear approximation to |𝜙 𝑗+1⟩, with error also (𝛿𝑡/𝜎)2. Keeping only

the leading terms, notice that the Taylor remainder error is subdominant. Altogether,

𝑒𝑖Δ𝛿𝑡 |𝜙 𝑗 ⟩ = |𝜙 𝑗+1⟩ + 𝑂

(cid:16)

(𝛿𝑡/𝜎)2 + √︁𝛿𝑡/𝜎𝑒−(𝑇/2𝜎)2 (cid:17)

.

(4.50)

So far, we’ve proved the result for 𝑚 = 1. The full result follows by noting that 𝑒𝑖Δ𝑚𝛿𝑡 = (𝑒𝑖Δ𝛿𝑡)𝑚

and taking, as upper bound, 𝑚 times the error of a single step.

□

We note that the error in Δ generating translations comes from two sources: the discretization

at small scales and the boundary effects at large scales. We might name these, in the language of

the lattice field theory, ultraviolet and infrared truncation effects, respectively.

Our next intermediate result will be concerned with the time evolution of the system under

𝐶 (𝐻) controlled on the Gaussian state |𝜙 𝑗 ⟩. We want the result to be, approximately, an evolution

under 𝐻 (𝑡 𝑗 ) on the main register of interest. In what follows, it will be convenient to take 𝜏 := 𝑇/𝑁 𝑝

as the time duration of a larger subdivision of steps.

88

Lemma 4.3.4. Let 𝐻 : [0, 𝑇] → Herm(H ) be a bounded differentiable function with bounded

derivative. For any 𝜂 ∈ R, we have

𝑒−𝑖𝐶 (𝐻)𝜂 |𝜓⟩|𝜙 𝑗 ⟩ = 𝑒−𝑖𝐻 (𝑡 𝑗 )𝜂 |𝜓⟩|𝜙 𝑗 ⟩

(cid:18)

+ 𝑂

𝜂𝜏 max
𝑡∈[0,𝑇]

∥ (cid:164)𝐻 (𝑡) ∥ + (1 + 𝜂 max
𝑡∈[0,𝑇]

∥𝐻 (𝑡) ∥)𝑒−𝜏2/4𝜎2 (cid:19)

where 𝜏 := 𝑇/𝑁 𝑝.

Proof. We begin by grouping the terms of 𝐶 (𝐻) into two chunks: one with significant overlap with

the Gaussian, the other with small overlap. Specifically, we take 𝐶 (𝐻) = 𝐻av + 𝐻⊥, with

𝐻av : =

𝑗+𝑁𝑞/2−1
∑︁

𝑘= 𝑗−𝑁𝑞/2

𝐻𝑘 ⊗ |𝑘⟩⟨𝑘 |

𝐻⊥ : = 𝐶 (𝐻) − 𝐻av.

(4.51)

Because 𝐻av and 𝐻⊥ commute, we can Trotterize with no error.

𝑒−𝑖𝐶 (𝐻)𝜂 |𝜓⟩|𝜙 𝑗 ⟩ = 𝑒−𝑖𝐻⊥𝜂𝑒−𝑖𝐻av𝜂 |𝜓⟩|𝜙 𝑗 ⟩

(4.52)

We will show that the 𝐻av term gives approximately 𝐻 (𝑡 𝑗 ), while 𝐻⊥ acts as approximately the

identity (with the right parameter values).

First, consider 𝑒−𝑖𝐻av𝜂. Define 𝑃 𝑗 = (cid:205)𝑘 |𝑘⟩⟨𝑘 | as the projector onto the clock states on which

𝐻av has support (𝑘 ∈ Z ∩ [ 𝑗 − 𝑁𝑞/2, 𝑗 + 𝑁𝑞/2 − 1]). We have

∥𝑒−𝑖𝐻av𝜂 − 𝑒−𝑖𝐻 𝑗 ⊗𝑃 𝑗 𝜂 ∥ ≤ 𝜂∥𝐻av − 𝐻 𝑗 ⊗ 𝑃 𝑗 ∥.

(4.53)

Meanwhile,

(cid:13)
(cid:13)𝐻av − 𝐻 𝑗 ⊗ 𝑃 𝑗

(cid:13)
(cid:13) =

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑗+𝑁𝑞/2−1
∑︁

𝑘= 𝑗−𝑁𝑞/2

(𝐻𝑘 − 𝐻 𝑗 ) ⊗ |𝑘⟩⟨𝑘 |

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

= max
𝑘

∥𝐻𝑘 − 𝐻 𝑗 ∥.

(4.54)

By a simple Taylor bound, ∥𝐻𝑘 − 𝐻 𝑗 ∥ ≤ (𝜏/2) max𝑡 ∥ (cid:164)𝐻 (𝑡) ∥, were the max is over [𝑡𝑘 , 𝑡 𝑗 ] (taking

the appropriate ordering of 𝑡 𝑗 , 𝑡𝑘 if needed). We can therefore say

∥𝑒−𝑖𝐻av𝜂 − 𝑒−𝑖𝐻 𝑗 ⊗𝑃 𝑗 𝜂 ∥ ≤ 𝜂𝜏 max
𝑡∈[0,𝑇]

∥ (cid:164)𝐻 ∥

(4.55)

89

so that, up to this error, we can replace a simulation by 𝐻av with 𝐻 𝑗 ⊗ 𝑃0. Moving on to this

situation, we have

𝑒−𝑖𝐻 𝑗 ⊗𝑃0𝜂 |𝜓⟩ ⊗ |𝜙 𝑗 ⟩ = 𝑒−𝑖𝐻 𝑗 𝜂 |𝜓⟩𝑃 𝑗 |𝜙 𝑗 ⟩ + |𝜓⟩(𝐼 − 𝑃 𝑗 )|𝜓 𝑗 ⟩.

(4.56)

Thinking of 𝜎 < 𝜏 and taking 𝜏/𝜎 increasing, we have 𝑃0|𝜙 𝑗 ⟩ = |𝜙 𝑗 ⟩ + 𝑂

(cid:16)

𝑒−𝜏2/4𝜎2 (cid:17)

. Thus,

𝑒−𝑖𝐻 𝑗 ⊗𝑃0𝜂 |𝜓⟩|𝜙 𝑗 ⟩ = 𝑒−𝑖𝐻 𝑗 𝜂 |𝜓⟩|𝜙 𝑗 ⟩ + 𝑂

(cid:18)

𝜂𝜏 max
𝑡∈[0,𝑇]

∥ (cid:164)𝐻 ∥ + 𝑒−𝜏2/4𝜎2 (cid:19)

.

(4.57)

For the remainder of the proof, take |𝜓′⟩ = 𝑒−𝑖𝐻 𝑗 𝜂 |𝜓⟩ for notational convenience. We now

consider the action of 𝐻⊥ on the remaining state, which we anticipate to be small. First,

(cid:13)𝑒−𝑖𝐻⊥𝜂 |𝜓′⟩|𝜙 𝑗 ⟩ − |𝜓′⟩|𝜙 𝑗 ⟩(cid:13)
(cid:13)

(cid:13) ≤ 𝜂∥𝐻⊥|𝜓′⟩|𝜙 𝑗 ⟩∥.

(4.58)

Let J be an index set for all the time steps included in the summation 𝐻av. We have
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
𝑘∉J
√︄

∥𝐻⊥|𝜓′⟩|𝜙 𝑗 ⟩∥ =

𝐻𝑘 |𝜓′⟩|𝑘⟩⟨𝑘 |𝜙 𝑗 ⟩

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

∑︁

≤

∑︁

𝑘∉J

1
N

𝑒−2|𝑡 𝑗 −𝑡𝑘 |2

𝑐/𝜎2 ∥𝐻𝑘 ∥2.

Employing a Hölder inequality on the inner product, followed by Lemma 4.3.2,

√︄

∑︁

𝑘∉J

1
N

𝑒−2|𝑡 𝑗 −𝑡𝑘 |2

𝑐/𝜎2 ∥𝐻𝑘 ∥2 ≤ max
𝑘∉J

∥𝐻𝑘 ∥

𝑒−|𝑡 𝑗 −𝑡𝑘 |2

𝑐/𝜎2

N

∑︁

𝑘∉J

∥𝐻 (𝑡) ∥(𝛿𝑡/𝜎)

max
𝑡

∈ 𝑂 (cid:169)
(cid:173)
(cid:171)

∞
∑︁

𝑘=𝑁𝑞/2

.

𝑒−𝑘 2𝛿𝑡2/𝜎2(cid:170)
(cid:174)
(cid:172)

(4.59)

(4.60)

Following a similar procedure to before, we convert to an error function erf and take an exponential

upper bound. Doing so gives

∥𝐻⊥|𝜓′⟩|𝜙 𝑗 ⟩∥ ∈ 𝑂

max
𝑡∈[0,𝑇]
𝜂 max𝑡 ∥𝐻 (𝑡) ∥𝑒−(𝜏/2𝜎)2 (cid:17)
Combining the errors together, we take the widest exponential 𝑒−𝜏2/4𝜎2

Thus, 𝑒−𝑖𝐻⊥𝜂 acts trivially on this state up to 𝑂

(cid:16)

.

.

(4.61)

as a simple upper bound

(cid:18)

∥𝐻 (𝑡) ∥𝑒−𝜏2/2𝜎2 (cid:19)

for all exponentials that appear. Putting all the error sources together gets us the result of the

Lemma statement.

□

90

With the previous two lemmas, we have the ingredients needed for a clock space simulation:

controlled operations and time shifts. We combine them to show that our clock space indeed

encodes time dependent dynamics.

Theorem 4.3.5. Let 𝐻 [0, 𝑇] → Herm(H ) be a time dependent Hamiltonian on a finite dimensional

vector space H , such that 𝐻 (𝑡) as a function is bounded and differentiable with bounded derivative.

Then, the clock Hamiltonian, with Gaussian input |𝜙0⟩, approximately applies the time evolution

operator 𝑈 (𝑇, 0) to an initial state |𝜓0⟩ ∈ H . Precisely,

𝑒−𝑖𝐻𝑐𝑇 |𝜓0⟩|𝜙0⟩ = (𝑈 (𝑇, 0)|𝜓0⟩) |𝜙0⟩

+ 𝑂

(cid:18)
𝑇 𝛿𝑡/𝜎2 + √︁𝑁𝑐𝑇/𝜎𝑒−𝑇 2/4𝜎2 + max

𝑡

∥ (cid:164)𝐻 ∥

𝑇 2
𝑁 𝑝

+ 𝑒−𝜏2/4𝜎2 (𝑁 𝑝 + max

𝑡

(cid:19)

.

∥𝐻 ∥𝑇)

Proof. Let 𝜏 := 𝑇/𝑁 𝑝. We begin with a first-order Trotterization of 𝐻𝑐 into 𝑁 𝑝 steps.

𝑒−𝑖𝐻𝑐𝑇 =

(cid:16)

𝑒𝑖Δ𝜏𝑒−𝑖𝐶 (𝐻)𝜏(cid:17) 𝑁 𝑝

(cid:18)

+ 𝑂

∥ (cid:164)𝐻 (𝑡) ∥

max
𝑡∈[0,𝑇]

(cid:19)

𝑇 2
𝑁 𝑝

(4.62)

With initial state |𝜓0⟩|𝜙0⟩, combining Lemmas 4.3.4 and 4.3.3 gives the following error for a single

Trotter step.

𝑒𝑖Δ𝜏𝑒−𝑖𝐶 (𝐻)𝜏 |𝜓0⟩|𝜙0⟩ = 𝑒−𝑖𝐻0𝜏 |𝜓0⟩|𝜙𝑁𝑞 ⟩

(cid:18)

+ 𝑂

𝜏𝛿𝑡/𝜎2 +

√︃

𝑁𝑞𝜏/𝜎𝑒−𝑇 2/4𝜎2 + 𝜏2 max

𝑡

∥ (cid:164)𝐻 ∥ + (1 + 𝜏 max

𝑡

∥𝐻 ∥)𝑒−𝜏2/4𝜎2 (cid:19)

(4.63)

Thus, after all 𝑁 𝑝 steps, we can multiply the single step error above to get an upper bound of

(cid:16)

𝑒𝑖Δ𝜏𝑒−𝑖𝐶 (𝐻)𝜏(cid:17) 𝑁 𝑝

|𝜓0⟩|𝜙0⟩ = 𝑒−𝑖𝐻𝑁𝑞 ( 𝑁𝑝 −1) 𝜏 . . . 𝑒−𝑖𝐻𝑁𝑞 𝜏𝑒−𝑖𝐻0𝜏 |𝜓0⟩|𝜙0⟩

+ 𝑂

(cid:18)
𝑇 𝛿𝑡/𝜎2 + √︁𝑁𝑐𝑇/𝜎𝑒−𝑇 2/4𝜎2 + max

𝑡

∥ (cid:164)𝐻 ∥

𝑇 2
𝑁 𝑝

+ 𝑒−𝜏2/4𝜎2 (𝑁 𝑝 + max

𝑡

(cid:19)

.

∥𝐻 ∥𝑇)

(4.64)

The right side, without the error, is a 1st order Suzuki Trotter splitting, which approximates 𝑈 (𝑇, 0)

to order max𝑡∈[0,𝑇] ∥𝐻 (𝑡)∥𝑇 2/𝑁 𝑝. This can be absorbed into the third term of the big-𝑂. This gives

the result stated in the Theorem.

□

With this result in hand, we now show that the parameters (𝑁 𝑝, 𝑁𝑞, 𝜎) of the clock can be

chosen such that any desired degree of approximation to 𝑈 (𝑇, 0) can be achieved.

91

Theorem 4.3.6. In the context of the previous theorem, for any 𝜖 ∈ R+, there exists clock parameters

(𝑁 𝑝, 𝑁𝑞, 𝜎) such that

with (𝑁 𝑝, 𝑁𝑞) scaling as

(cid:13)𝑒−𝑖𝐻𝑐𝑇 |𝜓0⟩|𝜙0⟩ − 𝑈 (𝑇, 0)|𝜓0⟩|𝜙0⟩(cid:13)
(cid:13)

(cid:13) < 𝜖

(cid:18)

𝑁 𝑝 ∈ Θ

∥ (cid:164)𝐻 ∥

max
𝑡∈[0,𝑇]

(cid:19)

𝑇 2
𝜖

, 𝑁𝑞 ∈ Θ

(cid:18) max𝑡 ∥ (cid:164)𝐻 ∥𝑇 2
𝜖 2

(cid:19)

𝑥2

, 𝜎 ∈ Θ

(cid:18)

𝜖
max𝑡 ∥ (cid:164)𝐻 ∥𝑇𝑥

(cid:19)

.

Here,

√︄

𝑥 :=

log

(cid:19)

(cid:18) Γ 𝑇
𝜖

(cid:26)

Γ := max

max
𝑡∈[0,𝑇]

∥ (cid:164)𝐻 ∥𝑇, 𝜖 max
𝑡∈[0,𝑇]

(cid:27)

.

∥𝐻 ∥

In particular, there exists a sequence (𝑁 𝑝 ( 𝑗), 𝑁𝑞 ( 𝑗), 𝜎( 𝑗)) of clock space parameters, such that

Tr𝑐 (𝑒−𝑖𝐻𝑐𝑇 |𝜓0⟩|𝜙0⟩) = 𝑈 (𝑇, 0)|𝜓0⟩

lim
𝑗→∞

where Tr𝑐 is a partial trace over the clock register, and Tr𝑐 (|Ψ⟩) ≡ Tr𝑐 (|Ψ⟩⟨Ψ|).

Proof. To ensure a total error within 𝜖 is achievable, it suffices to ensure that each of the five terms

constituting the error in Theorem 4.3.5 is within 𝑂 (𝜖) independently. From the onset, we will
choose 𝑁 𝑝 ∈ Θ (cid:0)max𝑡∈[0,𝑇] ∥ (cid:164)𝐻 ∥𝑇 2/𝜖 (cid:1) to satisfy the third term.

We next move to understand the necessary 𝜎 scaling. We parametrize it as

𝜎 = 𝜏/𝑥

(4.65)

with the hope that 𝑥 can be chosen to increase slowly (i.e., that the Gaussian states have width only

slightly smaller than the Trotter step size). For this, we focus on the last two terms, since they have

no 𝑁𝑞 dependence (which will set the smallest scales). We seek

max𝑡 ∥ (cid:164)𝐻 ∥𝑇 2
𝜖

𝑒−𝑥2/4 ∈ 𝑂 (𝜖),

∥𝐻 ∥𝑇 𝑒−𝑥2/4 ∈ 𝑂 (𝜖)

max
𝑡

(4.66)

which can be satisfied provided that 𝑥 is asymptotically lower bounded as
(cid:27)(cid:19)

(cid:18)

𝑥2 ∈ Ω

log max

(cid:26) max𝑡 ∥ (cid:164)𝐻 ∥𝑇 2
𝜖 2

, max𝑡 ∥𝐻 ∥𝑇
𝜖

(4.67)

= Ω (log(Γ 𝑇/𝜖)) .

92

This sets the scaling for 𝜎.

We move next to the first term to fix 𝑁𝑞, since the 2nd term is expected to be quite small. We

require 𝑇 𝛿𝑡/𝜎2 ∈ 𝑂 (𝜖), which is equivalent to

𝑁 𝑝𝑥2
𝑁𝑞

∈ 𝑂 (𝜖).

(4.68)

Therefore, there exists an 𝑁𝑞 ∈ Θ (cid:0)𝑁 𝑝𝑥2/𝜖 (cid:1), satisfying the bound. All that remains is the second

term, whose contribution can be easily shown to be subdominant compared to the other sources.

Therefore, the choice of parameter scaling suffice to achieve the desired error 𝜖.

We have shown that any desired precision 𝜖 for dynamical simulation can be accommodated

by appropriate choice of clock space parameters. Taking a sequence 𝜖 𝑗 → 0, we see there exists a

sequence of clock space evolutions whose limit, restricted to the main register, is 𝑈 (𝑇, 0).

□

The asymptotic scalings provided in the above theorem will be important in our discussion of

qubitization, where we will use them to derive a query complexity.

4.4 Time Dependent Qubitization

In the previous section, we developed a clock space construction which encoded a time de-

pendent Hamiltonian as a time independent one on an augmented, finite-dimensional space. The

removal of time-ordering using a clock register opens the door for quantum algorithms for time

independent Hamiltonian simulation to simulate the full clock-system dynamics directly. In par-

ticular, qubitization is an asymptotically optimal [88] simulation method that can only be applied

to time independent 𝐻. In this section, we propose the simulation of time dependent Hamiltoni-

ans through qubitization using finite clock registers. To be concrete, we will work with an input

model in which 𝐻 (𝑡) is a linear combination of fixed unitaries with time-varying coefficients. This

describes, for example, Pauli matrices on 𝑛 qubits with fluctuating coefficients.

4.4.1 Pseudo-Algorithm

We take 𝑛𝑐 qubits to provide a clock register of size 𝑁𝑐 = 2𝑛𝑐 , which are included with the main

register of interest. The initial state |𝜓0⟩ |𝜙0⟩ must be prepared on this joint register. We take |𝜓0⟩

of the main register as given, since this is necessarily application dependent. We must, however,

93

prepare a Gaussian state |𝜙0⟩ on the clock register of 𝑛𝑐 qubits with width 𝜎. Unsurprisingly, much

effort has been devoted to this task. [80, 110, 109, 71, 57]. For our purposes, we will simply refer

to the approach by Kitaev and Webb [79, 10] as efficient enough for our purposes. The Gaussian

will nonnegligible support over 𝑂 (𝑁𝑞) clock states, and their algorithm scales polynomially in the

number of qubits 𝑛𝑞 = log 𝑁𝑞 over the Gaussian. This cost is negligible compared to the simulation

costs that we are about to discuss.

Once the initial state is prepared, we employ qubitization to approximate 𝑒−𝑖𝐻𝑐𝑇 on the full

register. Given 𝐻 (𝑡) in LCU form, we need to express 𝐻𝑐 in LCU form as well, which is not

immediate. This is done through several applications of the Signature Matrix Decomposition. We

also truncate Δ at high frequencies to reduce computational cost, with little loss in accuracy. Details

of the LCU decomposition are provided in the next subsection.

Once 𝐻𝑐 is in LCU form, select SEL and prepare PREP circuits may be constructed to block

encode 𝐻𝑐 as

𝐻𝑐/∥𝑐∥1 = (⟨0| PREP† ⊗ 𝐼)SEL(PREP |0⟩ ⊗ 𝐼)

(4.69)

where ∥𝑐∥1 is the one-norm of the LCU coefficients. Standard qubitization can now be done on

this block encoded Hamiltonian [88]. The PREP circuit must create a "quasi-uniform" distribution

over some number 𝑁 of states, in the sense that, on the LCU auxiliary register,

|PREP⟩ =

𝐾−1
∑︁

√

𝑗=1

𝛿 | 𝑗⟩ +

𝑁
∑︁

√

𝑗=𝐾

𝛿′ | 𝑗⟩

(4.70)

with 𝛿, 𝛿′, 𝐾 and 𝑁 determined by parameters of simulation. Meanwhile the SEL circuit will need

to apply controlled 𝑈𝑖 operations, where 𝑈𝑖 is a unitary in the 𝐻 (𝑡) decomposition, and controlled

signature matrices. These second operations can be done with classical comparator circuits. Each

SEL will also require a Quantum Fourier Transform and its inverse on the clock register.

4.4.2 Block Encoding

To make progress, it seems 𝐻 (𝑡) itself should be expressible neatly in terms of unitaries. Thus,

we assume 𝐻 (𝑡) is of the form

𝐻 (𝑡) =

𝛼𝑖 (𝑡)𝑈𝑖

𝐿
∑︁

𝑖=1

94

(4.71)

where 𝑈 𝑗 are Hermitian and unitary (e.g., 𝑛-qubit signed Pauli operators) and 𝛼 𝑗 (𝑡) are nonnegative

real-valued functions on [0, 𝑇]. When we discretize, the coefficients 𝛼𝑖 𝑗 ≡ 𝛼𝑖 (𝑡 𝑗 ) will be particularly

important. Expanding out 𝐶 (𝐻) from equation (4.27) using (4.71),

𝐶 (𝐻) =

𝑁𝑐−1
∑︁

(cid:32)𝐿−1
∑︁

(cid:33)

𝛼𝑖 𝑗𝑈𝑖

⊗ | 𝑗⟩⟨ 𝑗 |

𝑖=0

𝑈𝑖 ⊗ 𝐷𝑖

𝑗=0
𝐿−1
∑︁

𝑖=0

=

𝐷𝑖 :=

𝑁𝑐−1
∑︁

𝑗=0

𝛼𝑖 𝑗 | 𝑗⟩⟨ 𝑗 |

(4.72)

(4.73)

where

is a diagonal operator on the clock register. There is a general technique for LCU constructions

of diagonal, or easily diagonalized, operators via a Signature Matrix Decomposition, which we

digress to discuss.

4.4.3

Interlude: Signature Matrix Decomposition or "Alternating Sign Trick"

In our manipulations of 𝐻𝑐, we have come across the problem of expressing a diagonal,

Hermitian matrix 𝐷 in LCU form. We will handle this in the present section. Although I did not

invent this technique, I had a sufficiently hard time finding a clear reference to it in the literature,

such that I felt an overview would be appropriate and possibly helpful to future researchers.

There are many unitary bases that exist, but in our case we have quite stringent requirements.

We ultimately want our decomposition to consist of Hermitian operations as well, meaning they

should be reflection operators (𝑈2 = 𝐼). Moreover, it is sensible to look for diagonal unitaries,

because our operator 𝐷 is diagonal. These requirements alone enforce that our unitaries are, in

fact, signature matrices: 𝑈 = diag(𝜆1, 𝜆2, . . . , 𝜆𝑛) where 𝜆 𝑗 = ±1.

For the moment, let’s imagine the entries of 𝐷 are all positive integers, and we allow ourselves

to add unitaries only in integer amounts. Think of each entry as a bucket of size 𝜆 𝑗 . We want to

add to the bucket, and we can only do so in units of +1 (by unitarity) or −1. Each time we add a

unitary, say, the identity, we are adding a unit to each bucket. Some of the buckets will fill up faster

than others because they are smaller, but we are not allowed to stop adding, per se. We can only

95

add or remove one unit, not zero, by unitarity. The next best thing we can do is, while the other

buckets are getting filled, add and remove 1 unit in alternating sequence. We need to do this until

the largest bucket has been filled, at which point we can stop.

Let’s put this all more formally. We want a sequence of unitaries that keep track of whether

the entries are above, or below, the number 𝑘 of additions that have already occurred. To this end,

define

𝑈𝑘 :=

𝑛
∑︁

𝑗=1

(−1) 𝑘 [𝑘 >𝜆 𝑗 ] | 𝑗⟩⟨ 𝑗 |

(4.74)

where [𝑃] is the boolean function for proposition 𝑃 assigning 1 to true, 0 to false. We see that, for

𝑘 even, 𝑈𝑘 = 𝐼 is the identity operator. while for odd 𝑘 𝑈𝑘 has eigenvalue −1 whenever 𝑗 is such

that 𝑘 > 𝜆 𝑗 . Then, if we take a sum "until the largest bucket" ∥𝐷 ∥ has been filled, we should obtain

𝐷. In fact,

∥𝐷 ∥
∑︁

𝑘=1

𝑈𝑘 =

𝑛
∑︁

𝑗=1

| 𝑗⟩⟨ 𝑗 | =

𝑛
∑︁

𝑗=1

| 𝑗⟩⟨ 𝑗 |

∥𝐷 ∥
∑︁

𝑘=1

(−1) 𝑘 [𝑘 >𝜆 𝑗 ]

and the inner sum can be written as

𝜆 𝑗
∑︁

𝑘=1

1 +

∥𝐷 ∥
∑︁

𝑘=𝜆 𝑗 +1

(−1) 𝑘 = 𝜆 𝑗 + 𝜖

(4.75)

(4.76)

where 𝜖 ∈ {−1, 0, 1} is an 𝑂 (1)-error. We will see shortly how to boost the precision, but first we

note that, generalizing to positive real-valued entries, the same procedure for 𝑈𝑘 (taking ⌈∥𝐷 ∥⌉ as

the upper sum limit) generates ⌊𝜆 𝑗 ⌋ on the entries to accuracy ±1 at worst. Thus, for real values

the error is less than 2, which is still 𝑂 (1).

This might not seem like a good approximation, especially when 𝜆 𝑗 is small. But we can

artificially increase the size of 𝜆 𝑗 by performing the same procedure on 𝐷/𝛿 for suitably small

𝛿 > 0, then multiplying by 𝛿. Let 𝐿𝛿 := ⌈∥𝐷 ∥/𝛿⌉. Then

so

𝐷/𝛿 =

𝐿 𝛿
∑︁

𝑘=1

𝑈𝑘 + 𝑂 (1)

𝐷 =

𝐿 𝛿
∑︁

𝑘=1

𝛿𝑈𝑘 + 𝑂 (𝛿)

96

(4.77)

(4.78)

with 𝑈𝑘 the same as (4.74) but with the replacement 𝜆 𝑗 → 𝜆 𝑗 /𝛿. We’ve succeeded at expressing

𝐷 in LCU form to accuracy 𝑂 (𝛿) using 𝐿𝛿 terms.

We still haven’t handled negative eigenvalues. This is accomplished by adding the appropriate

sign. Altogether, the 𝐿𝛿 matrices

𝑈𝑘 :=

𝑛
∑︁

𝑗=1

sgn(𝜆 𝑗 ) (−1) 𝑘 [𝑘 >|𝜆 𝑗 |/𝛿]

(4.79)

are sufficient to approximate 𝐷 to within 2𝛿 in each entry.

Assuming the eigenvalues 𝜆 𝑗 are known and classically computable, unitaries such as (4.79)

can be implemented on a quantum computer using comparator circuits. Observe that this procedure

can also generate an LCU type expansion, more generally, when the Hermitian operator 𝐻 is easily

diagonalizable. Moreover, the fact that the coefficients in the LCU are the same allows for simple

implementation in a select-and-prepare block encoding.

4.4.4 Block Encoding (cont.)

Let Λ𝑖 (𝛿) ≡ ⌈max 𝑗 |𝛼𝑖 𝑗 |/𝛿⌉. Using a signature matrix decomposition, we can write

for 𝛿 > 0, where

𝐷𝑖 =

Λ𝑖 (𝛿)
∑︁

𝑘=1

𝛿𝑆𝑖𝑘 (𝛿) + 𝑂 (𝛿)

𝑆𝑖𝑘 (𝛿) =

𝑁𝑐−1
∑︁

𝑗=0

(−1) 𝑘 [𝑘 >𝛼𝑖 𝑗 /𝛿] | 𝑗⟩⟨ 𝑗 |

(4.80)

(4.81)

and [𝑃] is the Boolean function for proposition 𝑃, with [True] = 1 and [False] = 0. Thus, we

obtain an LCU decomposition of 𝐶 (𝐻) as

𝐶 (𝐻) = 𝛿

𝐿
∑︁

Λ𝑖 (𝛿)
∑︁

𝑖=1

𝑘=1

𝑈𝑖 ⊗ 𝑆𝑖𝑘 (𝛿) + 𝑂 (𝐿𝛿).

(4.82)

The prepare circuit PREP is simple enough because the linear combination is uniform. Therefore,

it can be accomplished using a Hadamard gate on each of

(cid:32)

𝑛𝐶 (𝐻) ∈ 𝑂

log

𝐿
∑︁

max
𝑗

(cid:33)

|𝛼𝑖 𝑗 |/𝛿

(4.83)

𝑖=0
auxiliary qubits needed for a binary encoding. The unitaries 𝑈𝑖 ⊗ 𝑆𝑖𝑘 (𝛿) can be selected using two

different SEL circuits: one for the original 𝑈𝑖 (presumed available to us) and one for the signature

97

matrices 𝑆𝑖𝑘 (𝛿). These unitaries can be constructed using classical comparator circuits provided

that each 𝛼𝑖 𝑗 is computable.

We turn our attention now to Δ. Although already in LCU form, the coefficient has size 2/𝛿𝑡

and is too large to be desirable. As discussed above, the problem stems from unnecessary high-

frequency modes, which we wish to truncate. We start by converting Δ to Fourier space, i.e.,

diagonalizing via the Quantum Fourier Transform. The result may be computed by diagonalizing

𝑈+, and is found to be

𝑁𝑐−1
∑︁

Δ = QFT

𝑁𝑐
𝑇 sin

(cid:18)

2𝜋

(cid:19)

𝑗
𝑁𝑐

| 𝑗⟩⟨ 𝑗 | QFT†

𝑗=0
𝑁𝑐/2−1
∑︁

𝑗=−𝑁𝑐/2

= QFT

𝑁𝑐
𝑇 sin

(cid:18)

2𝜋

(cid:19)

𝑗
𝑁𝑐

| 𝑗⟩⟨ 𝑗 | QFT†

(4.84)

where, in the second line, we define indices − 𝑗 = 𝑁𝑐 − 𝑗 for 𝑗 > 0 and write the diagonalized

Δ symmetrically about 𝑗 = 0. The benefit of this parametrization is that small | 𝑗 | correspond to

low-frequency modes, as we shall see. Let Δ𝐽 be Δ truncated at frequencies above those determined

by index 𝐽 ∈ [0, 𝑁𝑐/2] ∩ Z.

Δ𝐽 := QFT

𝐽
∑︁

𝑗=−𝐽

𝑁𝑐
𝑇 sin

(cid:18)

2𝜋

(cid:19)

𝑗
𝑁𝑐

| 𝑗⟩⟨ 𝑗 | QFT†

(4.85)

The error in a clock space evolution using Δ𝐽 rather than Δ is upper bounded by 𝑇 ∥Δ |𝜙0⟩−Δ𝐽 |𝜙0⟩ ∥,

which can be evaluated and upper bounded as

𝑇 ∥Δ |𝜙0⟩ − Δ𝐽 |𝜙0⟩ ∥ =

(cid:13)
(cid:13)
(cid:13)
(cid:13)

∑︁

𝑁𝑐 sin(2𝜋

𝑗
𝑁𝑐

)| 𝑗⟩⟨ 𝑗 |QFT† |𝜙0⟩

(cid:13)
(cid:13)
(cid:13)
(cid:13)

| 𝑗 |>𝐽
√︄ ∑︁

≤ 𝑁𝑐

|⟨ 𝑗 | QFT† |𝜙0⟩|2.

(4.86)

| 𝑗 |>𝐽

We thus desire a characterization of QFT† |𝜙0⟩, which we naturally expect to be another Gaussian

up to errors arising from the difference between discrete and continuous Fourier Transforms. This

analysis was performed in Appendix C of [113], and we adapt that work to our present situation.

As the reference shows, the error in each component 𝑗 arises from three sources:

1. Truncation of the time variable to 𝑂 (𝑇), which we denote 𝜖trunc.

98

2. Truncation of the frequency variable to 𝑂 (𝑁𝑐/𝑇) ("aliasing"), which we denote 𝜖alias.

3. Differences in normalizing in the continuum vs the discrete setting, which we denote 𝜖norm.

In our notation and setting, Rendon et al. [113] show that these errors satisfy the following

asymptotic bounds.

𝜖trunc ∈ 𝑂

𝜖alias ∈ 𝑂

𝜖norm ∈ 𝑂

(cid:19)

𝑒−Ω(𝑇 2/𝜎2)

𝑐 𝜎2/𝑇 2 (cid:19)

𝑒−Ω(𝑁 2

(cid:18)√︂ 𝜎
𝑇
(cid:18)√︂ 𝜎
𝑇
𝑒−Ω(𝑁𝑐)(cid:17)

(cid:16)

(4.87)

Let’s take these errors to all be 𝑂 (𝜖QFT), with the required 𝜖QFT to be determined. The results from

Theorem 16 and Appendix C of [113] imply that
(cid:32)√︂ 𝜋𝑁𝑐
N

QFT† |𝜙0⟩ =

𝑁𝑐/2−1
∑︁

𝑗=−𝑁𝑐/2

𝜎
𝑇

𝑒−(𝜋 𝑗 𝜎/𝑇)2 + 𝑂 (𝜖QFT)

(cid:33)

| 𝑗⟩ .

With this in hand, we return to (4.86). First,

|⟨ 𝑗 | QFT† |𝜙0⟩|2 =

𝜋𝑁𝑐
N

𝜎2
𝑇 2

𝑒−2(𝜋 𝑗 𝜎/𝑇)2 + 𝑂 (

√︂ 𝜎
𝑇

𝑒−(𝜋 𝑗 𝜎/𝑇)2𝜖QFT)

(4.88)

(4.89)

where we assume the error 𝜖QFT is smaller asymptotically than the amplitude itself, to be justified.

Taking the sum over high frequencies,

√︄ ∑︁

| 𝑗 |>𝐽

|⟨ 𝑗 | QFT† |𝜙0⟩|2 ∈ 𝑂

(cid:32)√︂ 𝑁𝑐
N

𝜎
𝑇

𝑒−Ω(𝐽2𝜎2/𝑇 2) +

√︂ 𝜎
𝑇

(cid:33)

𝑒−Ω(𝐽2𝜎2/𝑇 2)𝜖QFT

⊆ 𝑂

(cid:18)√︂ 𝜎
𝑇

𝑒−Ω(𝐽2𝜎2/𝑇 2) (1 + 𝜖QFT)

(cid:19)

.

(4.90)

We next observe that 𝜖QFT ∈ 𝑂 (1) by previous assumptions, and can now be removed. From (4.86),

we get the full simulation error by multiplying by 𝑁𝑐

𝜖𝐽 ∈ 𝑂

(cid:18)

𝑁𝑐

√︂ 𝜎
𝑇

𝑒−Ω(𝐽2𝜎2/𝑇 2)

(cid:19)

.

In order for 𝜖𝐽 ∈ 𝑂 (𝜖), we want the cutoff 𝐽 to satisfy

𝑒−𝐽2𝜎2/𝑇 2 ∈ 𝑂

(cid:33)

(cid:32)√︂ 𝑇
𝜎

𝜖
𝑁𝑐

99

(4.91)

(4.92)

which can be satisfied provided 𝐽 scales as

𝐽 ∈ Θ

(cid:18) 𝑇
𝜎

(log 𝜎/𝑇 + log 𝑁𝑐 + log 1/𝜖)

(cid:19)

⊆ ˜Θ(𝑇/𝜎).

(4.93)

Letting ˜Δ ≡ Δ𝐽 for this choice of 𝐽, we now switch to considering the simulation of ˜Δ. Let 𝛿′ > 0,

and let Γ(𝛿′) := ⌈(𝑁𝑐/𝑇 𝛿′) sin(2𝜋𝐽/𝑁𝑐)⌉. We have

where

𝐽
∑︁

𝑗=−𝐽

𝑁𝑐
𝑇 sin

(cid:18)

2𝜋

(cid:19)

𝑗
𝑁𝑐

| 𝑗⟩⟨ 𝑗 | = 𝛿′

Γ(𝛿′)
∑︁

ℓ=1

𝑆(Δ)
𝑘

(𝛿′) + 𝑂 (𝛿′)

𝐽
∑︁

𝑆(Δ)
𝑘

(𝛿′) :=

sgn( 𝑗) (−1) 𝑘 [𝑘 >(𝑁𝑐/𝑇 𝛿′) sin(2𝜋 𝑗/𝑁𝑐)] .

(4.94)

(4.95)

𝑗=−𝐽
Defining the unitary 𝑉ℓ (𝛿′) := QFT 𝑆(Δ)

ℓ

(𝛿′) QFT†, we have obtained an LCU decomposition of Δ.

The PREP circuit is, as with 𝐶 (𝐻), only a column of Hadamards on

𝑛Δ ∈ 𝑂 (log ((𝑁𝑐/𝑇 𝛿′) sin(2𝜋𝐽/𝑁𝑐))) ⊆ ˜𝑂

(cid:18)

log

(cid:19)

1
𝜎𝛿′

(4.96)

auxiliary qubits. Meanwhile the SEL circuit may be constructed as QFT SEL′ QFT†, where SEL′ is a
select circuit using the 𝑆(Δ)

signature matrices that can, as before, be implemented with comparator

ℓ

circuits that compute sine.

Combining with (4.82), we obtain an approximate LCU decomposition of the approximate

clock Hamiltonian ˜𝐻𝑐.

˜𝐻𝑐 = 𝛿

𝐿
∑︁

Λ𝑖 (𝛿)
∑︁

𝑖=1

𝑘=1

𝑈𝑖 ⊗ 𝑆𝑖𝑘 (𝛿) + 𝛿′

Γ(𝛿′)
∑︁

ℓ=1

𝐼 ⊗ 𝑉ℓ (𝛿′) + 𝑂 (𝜖/𝑇 + 𝐿𝛿 + 𝛿′)

(4.97)

To achieve an 𝜖-accurate simulation, we will require 𝛿 ∈ 𝑂 (𝜖/𝐿𝑇) and 𝛿′ ∈ 𝑂 (𝜖/𝑇). The 1-norm

100

∥𝑐∥1 of all of the coefficients is given by

𝐿−1
∑︁

∥𝑐∥1 = 𝛿

Λ𝑖 (𝛿) + 𝛿′Γ(𝛿′)

𝑖=0
(cid:32)𝐿−1
∑︁

𝑖=0

∈ 𝑂

|𝛼𝑖 𝑗 | +

max
𝑗

(cid:33)

𝑁𝑐
𝑇 sin(2𝜋𝐽/𝑁𝑐)

(cid:16)

(cid:16)

(cid:18)

⊆ 𝑂

⊆ ˜𝑂

⊆ ˜𝑂

∥𝛼∥rev

∥𝛼∥rev

(cid:17)

∞,1 + 𝐽/𝑇
∞,1 + 𝜎−1(cid:17)

∥𝛼∥rev

∞,1 +

max𝑡 ∥ (cid:164)𝐻 ∥𝑇
𝜖

(cid:19)

(4.98)

where ∥𝛼∥∞,1 ≡ (cid:205)𝐿−1

𝑖=0 max𝑡 |𝛼𝑖 (𝑡)| and ˜𝑂 suppresses multiplicative logarithmic factors. Thus, the

number of queries to SEL and PREP circuits in an LCU encoding scales as

(cid:18)

𝑄 ∈ ˜𝑂

∥𝛼∥rev
∞,1

𝑇 +

max𝑡 ∥ (cid:164)𝐻 ∥𝑇 2
𝜖

+

log 1/𝜖
log log 1/𝜖

(cid:19)

.

The number of auxiliary qubits needed for the clock register is

𝑛𝑐 = log 𝑁 𝑝 + log 𝑁𝑞 ∈ 𝑂

(cid:16)

log(max

𝑡

∥ (cid:164)𝐻 ∥𝑇 2) + log 1/𝜖

(cid:17)

while the number of auxiliary qubits needed for the LCU block encoding is given by

𝑛LCU = 𝑛𝐶 (𝐻) + 𝑛Δ
(cid:32)

∈ 𝑂

log

∥𝛼∥rev
∞,1
𝛿

+ log

(cid:33)

1
𝜎𝛿′

(cid:32)

⊆ 𝑂

log

𝑇

𝐿 ∥𝛼∥rev
∞,1
𝜖

+ log

(cid:33)

max𝑡 ∥ (cid:164)𝐻 ∥𝑇 2
𝜖 2

(cid:16)

⊆ 𝑂

log 𝐿 + log(∥𝛼∥rev
∞,1

𝑇) + log(max

𝑡

∥ (cid:164)𝐻 ∥𝑇 2) + log 1/𝜖

(cid:17)

(4.99)

(4.100)

(4.101)

for a total number of auxiliary qubits 𝑛 ∈ 𝑂 (𝑛LCU).

Improvements over 1st-order Trotter in query complexity, with our bounds, only appear with

very small time variations. We may be able to prove this through better characterizations in the

error stemming from 𝐻𝑐. We use Trotter bounds in those calculations, but we may need to be

smarter to avoid the limits seen here.

101

4.5 Discussion

In the circuit-based Hamiltonian simulation community, time dependent Hamiltonians are often

treated on separate footing from time independent ones. The main contribution of this project is

a new way of thinking about time dependent dynamics that allows us to replace time ordered

operator exponentials with ordinary operator exponentials acting on a higher-dimensional finite

space. We apply the discretized (𝑡, 𝑡′) trick and show that it encodes the time ordered exponential

for sufficiently large clock sizes. The clock space framework can also be used directly in quantum

simulation methods to extend the capabilities of certain quantum algorithms. Specifically, it can

be used to extend qubitization to time dependent systems. While in many circumstances it will be

more convenient to use a truncated Dyson series simulation method in preference to this approach,

our work shows how the use of discrete clock spaces used to construct new quantum simulation

algorithms that would otherwise be challenging.

Besides an LCU encoding, natural block encodings of 𝐻𝑐 may be possible. For example, a very

general input model for 𝐻 (𝑡) is to take it as a 𝑑-sparse matrix with query access to the nonzero

entries. This seems quite promising an avenue to take, because then 𝐻𝑐 = 𝐶 (𝐻) + Δ is 𝑑 + 2

sparse, and there is a natural way to query the entries of 𝐻𝑐. Hence, such a Hamiltonian should

immediately simulatable by qubitization (or other quantum walk methods). The trouble is that the

largest entry in absolute value ∥𝐻𝑐 ∥max of 𝐻𝑐 comes from Δ, which is of size 𝑁𝑐/2𝑇. This is too

large to yield an effective simulation algorithm. Of course, there is something odd about the need to

care for the operator norm ∥Δ∥, since the typical state being acted on is a Gaussian |𝜙 𝑗 ⟩. Thinking

of Δ in frequency space, modes of frequency Ω(𝜎−1) should not be relevant for Gaussian states of

width 𝑂 (𝜎) on the clock register. This suggests that a high-frequency truncation of Δ, say ˜Δ would

act approximately the same on the Gaussians while decreasing the norm. However, there is no

guarantee that the modified operator, ˜Δ, is sparse in the basis of clock times. Perhaps considering

a reduced clock Hamiltonian ˜𝐻𝑖 𝑗 = ⟨𝜙𝑖 |𝐻𝑐 |𝜙 𝑗 ⟩, with all small elements set to zero, would have the

sparseness conditions required, along with a subspace norm of ∥Δ∥𝜙 ∈ 𝑂 (𝜎−1). Investigating such

sparse encodings would make an interesting avenue for future work.

102

CHAPTER 5

MULTIPRODUCT FORMULAS FOR TIME DEPENDENT SIMULATION

One of the key developments of Chapter 4 is the construction of a discrete clock space which reduces

the computation of time dependent Hamiltonians to time independent ones. This reduction provides

a useful way to translate techniques and concepts between the time dependent and time independent

settings. In this chapter, we investigate one of these connections: a generalization of Multiproduct

Formulas (MPFs) for the time dependent setting based on product formula simulations of the clock

space. After arguing that such "time dependent MPFs" should form good approximations to the

time evolution operator 𝑈 for sufficiently smooth 𝐻 (𝑡), we propose an algorithm based on these

MPFs. We then provide a rigorous characterization of error in these formulas, and from this derive

a query complexity in a natural Hamiltonian input model. Numerical demonstrations are used to

validate the effectivess of time dependent MPFs at achieving high-accuracy simulations.

What we find for the properties of the MPF algorithm (and the qubitization algorithm) is

summarized in Table 5.1, where we also display other leading algorithms for time dependent

Hamiltonian simulation. Overall, the MPF algorithm has comparable performance to the Dyson

method, with strengths and weaknesses on both sides. For example, unlike the Dyson method, the

MPF simulation exhibits commutator scaling, meaning that the simulation is perfect for commuting

Hamiltonian terms and no time dependence. It also scales as the more favorable 𝐿1 norm of the

Hamiltonian rather than the maximum value at a given time, as the Dyson series does. On the other

hand, the Dyson series is not concerned with large derivatives, but only the size of 𝐻. Overall, time

dependent MPF simulation enlarges the collection of available tools for the future practitioner who

is looking for the right algorithm for their problem of interest.

This chapter is the subject of ongoing research, particularly with respect to the proof (or

disproof) of Conjecture 1. An early preprint has been posted [133] which will be updated as the

project reaches completion.

103

Method

Trotter [137]

Query Complexity Auxiliary Qubits
𝑂 (cid:0)𝐿 (∥Λ∥1)1+𝑜(1)/𝜖 𝑜(1)(cid:1)
𝑂 (∥𝛼∥2

0

1,1/𝜖)

QDrift [16]
Dyson [89, 75] 𝑂 (cid:0)∥𝛼∥1,∞𝑇 log(1/𝜖)(cid:1)
𝑇 + log 1/𝜖 (cid:1)

(cid:101)𝑂 (cid:0)∥𝛼∥rev
∞,1
∥ (cid:164)𝐻 ∥𝑇 2/𝜖 (cid:1)
+ max
𝑡

Qubitization*

MPF

(cid:101)𝑂 (cid:0)𝐿 ∥Λ∥1 log2(1/𝜖)(cid:1)

0
No
(cid:101)𝑂 (cid:0) log(∥ (cid:164)𝛼∥1,1/𝜖) + log(∥𝛼∥1,∞𝑇/𝜖)(cid:1) No
(cid:17)
𝑂 (cid:0) log

𝑇/𝜖

(cid:16)

𝐿 ∥𝛼∥rev
∞,1
∥ (cid:164)𝐻 ∥𝑇 2/𝜖

(cid:16)

(cid:17) (cid:1)

𝐿 max
𝑡

+ log
(cid:101)𝑂 (cid:0)log (cid:0)𝐿∥Λ∥1∥ (cid:164)𝛼∥∞,∞𝑇 2/𝜖 (cid:1)(cid:1)

No

Yes

CS?

Yes

Table 5.1 Summary of our results (green) and comparison to leading quantum simulation methods
for time dependent Hamiltonians. We assume that 𝐻 = (cid:205)𝐿
and real-valued 𝛼 𝑗 (𝑡). Λ is a positive, time dependent function with dimensions of 𝐻, and
quantifies the size of 𝛼 𝑗 and its derivatives (see Definition 3). ∥𝛼∥ 𝑝,𝑞 refers to a nested vector-𝑝
and functional-𝑞 norm for the coefficients 𝛼 = (𝛼 𝑗 ) 𝐿
reverse order. Commutator scaling (CS) here means the simulation error vanishes in the limit
where 𝐻 is time independent and [𝑈 𝑗 , 𝑈𝑘 ] = 0 for all 𝑗, 𝑘 ∈ [𝐿].

𝛼 𝑗 (𝑡)𝑈 𝑗 for Hermitian unitaries 𝑈 𝑗

𝑝,𝑞 indicates these are taken in the

𝑗=1, and ∥𝛼∥rev

𝑗=1

5.1

Introduction and Background

Multiproduct formulas (MPFs) are a generalization of the celebrated product formulas and

span two of the pillars of quantum simulation: product formula and LCU methods. The aim of

the MPF is to approximate the time evolution operator 𝑈 as a linear combination of lower-order

Trotter formulas, in such a way that higher order errors are cancelled [28, 24, 90]. They are,

fundamentally, nothing more than a Richardson extrapolation of a product formula P to Trotter

step size 𝑠 → 0. This summation is done to address the primary deficiency of product formulas:

the cost of constructing a high order product formula is exponentially large. This is not only true

of the well-known Suzuki-Trotter formulas, but any similar construction, due to the need to cancel

error terms that grow exponentially in the number of products considered. In contrast, since the

MPF is a sum of product formula approximations, the number of error terms of a given order does

not grow exponentially. This allows us to approximate the quantum dynamics using polynomially

many, rather than exponentially many, operator exponentials.

The current central result concerning the use of MPFs for time independent simulation is

104

provided by the following theorem of Low, Kliuchnikov, and Wiebe [90].

Theorem 5.1.1 (Time independent MPFs (Theorem 1 of [90])). Let 𝐻 be a bounded, time inde-

pendent Hamiltonian, and let 𝑈2(𝑡) be the 2nd-order Suzuki-Trotter formula for the time evolution
operator 𝑈 (𝑡) = 𝑒−𝑖𝐻𝑡. Let 𝑎 = (𝑎1, 𝑎2, . . . , 𝑎𝑚) ∈ R𝑚 and (cid:174)𝑘 = (𝑘1, 𝑘2, . . . , 𝑘𝑚) ∈ Z𝑚
choices of 𝑎 and (cid:174)𝑘 such that the multiproduct formula

+ . There exist

𝑈2,𝑚 (𝑡) :=

𝑚
∑︁

𝑗=1

𝑎 𝑗𝑈 𝑘 𝑗
2

(𝑡/𝑘 𝑗 )

is order 2𝑚 and satisfies

𝑘 𝑗 ∈ 𝑂 (𝑚2),

max
𝑗

∥𝑎∥1 ∈ 𝑂 (polylog(𝑚)).

As a caution, we remark that, despite notation, the MPF 𝑈2,𝑚 is not generally unitary for 𝑚 > 1,

though when suitably constructed it will approximate the unitary 𝑈 (hence be approximately

unitary). The proof of Theorem 5.1.1 may be found in [90], but at a high level, the MPF 𝑈2,𝑚

is a Richardson extrapolation of 𝑈2 with respect to the Trotter step size parameter 1/𝑘. Such an

extrapolation is possible for arbitrary 𝑚 because there exists an error series [18]

𝑈 𝑘
2 (𝑡/𝑘) − 𝑈 (𝑡) =

𝐸2 𝑗+1

𝑡2 𝑗+1
𝑘 2 𝑗

∞
∑︁

𝑗=1

(5.1)

with 𝐸2 𝑗+1 independent of 𝑘 (but not 𝑡 generically). The existence of this series suffices for a

1/𝑘 → 0 Richardson extrapolation [119].

In particular, cancellation occurs for coefficients 𝑎 𝑗

satisfying the following Vandermonde linear system.

1

−2

𝑘1
...

−2𝑚+2

𝑘1

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

· · ·

· · ·
. . .

· · ·

1

−2

𝑘𝑚
...

−2𝑚+2

𝑘𝑚

𝑎1

𝑎2
...

𝑎 𝑀

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

=

1
(cid:170)
(cid:174)
(cid:174)
0
(cid:174)
(cid:174)
...
(cid:174)
(cid:174)
(cid:174)
(cid:174)
0
(cid:172)

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(5.2)

Though the matrix is ill-conditioned, this is irrelevant to numerical stability, as the inverse Van-

dermonde matrix admits an analytic solution that may be reasoned from the theory of polynomial

interpolation. What matters for our application is the one-norm ∥𝑎∥1 of the coefficients, which

105

serves as our "condition number" because of how it amplifies small errors in the base formula

𝑈2 [90]. The content of Theorem 5.1.1 is that Trotter steps (cid:174)𝑘 may be chosen such that ∥𝑎∥1 is not

too large. For time-ordered 𝑈, the analysis of [18] does not carry over, although reasonable "time

dependent" MPFs can be defined heuristically. One of our motivations in constructing a clock

space is to be able to eliminate time ordering and, in this setting, show these formulas work.

As discussed in [90], specific choices of 𝑘 𝑗 can be found numerically to minimize ∥𝑎∥1, and this

may be the best approach in practice. However, for our analytical results it will be most appropriate

to utilize the specific 𝑘 𝑗 chosen in their constructive proof of well-conditioned MPFs. Thus, for all

results we will take the powers 𝑘 𝑗 as follows.

𝑘 𝑗 =

(cid:38) √

8𝑚
𝜋

(cid:12)
(cid:12)
(cid:12)
(cid:12)

sin

(cid:18) 𝜋(2 𝑗 − 1)
8𝑚

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

−1(cid:39)

,

𝑗 = 1, . . . , 𝑚

(5.3)

We will use these same coefficients even in the time dependent MPFs to be introduced in the

subsequent section. For error analysis, it will be useful to have simple, concrete bounds on 𝑘 𝑗 . We

can achieve this by noting that sin(𝑥) ≤ 𝑥 and sin(𝑥) ≥ 4𝑥/5 for 𝑥 ∈ [0, 1]. This gives the lower

bound

and the upper bound

𝑘 𝑗 ≥

(cid:25)

(cid:24) 83/2𝑚2
𝜋2(2 𝑗 − 1)

(cid:24)

≥

83/2𝑚2
𝜋2(2𝑚 − 1)

(cid:25)

>

√

128𝑚
𝜋2

> 𝑚

𝑘 𝑗 ≤

(cid:38)

√

8𝑚2
5 × 8
4(2 𝑗 − 1)𝜋2

(cid:39)

(cid:38)

≤

√

8𝑚2

(cid:39)

5 × 8

4𝜋2

< 3𝑚2.

Note the consistency of (5.5) with the big-𝑂 scaling of Theorem 5.1.1.

5.2 Definition and Effectiveness

(5.4)

(5.5)

Multiproduct formulas have already been considered extensively in the Hamiltonian simulation

community [24, 45, 141], however, they have yet to be seriously considered for use in time dependent

Hamiltonian simulations. Because 𝑈 generally has time-ordering, the techniques used in [18]

involving Baker-Campbell-Hausdorff-type expansions do not carry over directly. An approach

based instead on the Magnus expansion might be expected to work in its place, but no subset of

terms in the expansion represents the exact evolution separated from error terms. Without this

106

generalization, MPFs cannot be applied to interaction picture algorithms as well as simulations of

physical systems that have intrinsic time dependence.

It is rather easy to propose a reasonable generalization of the MPFs of Theorem 5.1.1 that

would be expected to work well in the time dependent case. Simply replace the 𝑘 𝑗 th power with

a sequence of 𝑘 𝑗 unitaries at each time slice. We will present this definition shortly. First, as

a preliminary step, we want to clearly define a notion of approximation order of two-parameter

operator functions (such as the propagator 𝑈 (𝑡, 𝑡0)) that will suit our purposes.

Definition 1. For finite-dimensional H , let 𝐿 : [0, 𝑇]2 → 𝐿(H ). We say that 𝐿 𝑝 : [0, 𝑇]2 → 𝐿(H )

is a pth-order approximation to 𝐿 if, for all 𝑡 ∈ [0, 𝑇),

∥𝐿 (𝑡 + 𝜏, 𝑡) − 𝐿 𝑝 (𝑡 + 𝜏, 𝑡) ∥ ∈ 𝑂 (𝜏 𝑝+1)

where 𝜏 is taken asymptotically to 0.

Observe that this aligns with the regular notion of 𝑝th order formulas when considered as a

function of a single variable 𝜏 and fixed 𝑡. With 𝑝th order approximants defined, we now propose

a generalization of MPFs for two-parameter operators such as the general propagator 𝑈.

Definition 2 (Time Dependent Multiproduct Formulas). For finite dimensional H and 𝐿 : [0, 𝑇]2 →
+ , and 𝑎 ∈ R𝑚,

𝐿 (H ), let 𝐿 𝑝 : [0, 𝑇]2 → 𝐿(H ) be a 𝑝th-order formula for 𝐿. Given 𝑚 ∈ Z+, (cid:174)𝑘 ∈ Z𝑚

define the time dependent Multiproduct Formula 𝐿𝑚,𝑝 : [0, 𝑇]2 → 𝐿 (H ) to be

𝐿𝑚,𝑝 (𝑡, 𝑡0) :=

𝑚
∑︁

𝑗=1

𝑎 𝑗 𝐿 (𝑘 𝑗 )
𝑝

(𝑡, 𝑡0)

𝐿 (𝑘)

𝑝 (𝑡, 𝑡0) :=

𝑘−1
(cid:214)

ℓ=0

𝐿 𝑝 (𝑡ℓ+1, 𝑡ℓ)

where

and 𝑡ℓ = 𝑡0 + (𝑡 − 𝑡0)ℓ/𝑘.

The choice to take the 𝑡ℓ as equally spaced is not entirely coincidental, for the same reason that,

in the time independent setting, we take 𝑈 𝑘
2

(𝑡/𝑘) instead of, say,

𝑘
(cid:214)

𝑗=1

𝑈2(𝑠 𝑗 𝑡)

107

(5.6)

where 𝑠 = (𝑠1, . . . , 𝑠𝑘 ) is a probability vector. Taking a simple power of 𝑘 makes reasoning with

the BCH expansion possible.

While these definitions are likely applicable in more general contexts (such as classical time

dependent symplectic dynamics), our interest in simulation means we will consider 𝐿 = 𝑈 to be a

time evolution operator, satisfying all the corresponding properties. Moreover, we will assume 𝐻

is in the so-called linear combination of Hamiltonians (LCH) form

𝐻 =

𝐿
∑︁

𝑖=1

𝐻𝑖 (𝑡)

(5.7)

which is suitable for product formula simulations. Later we will make the assumption that the

exponentials of each 𝐻 𝑗 (𝑡) can be efficiently computed, a standard assumption. Because we utilize

the well-conditioning results of [90], we want the base formula to be 2nd order and symmetric.

Thus, we will take 𝐿 𝑝 = 𝑈2 to being the 2nd order symmetric midpoint formula

𝑈2(𝑡 + 𝜏, 𝑡) :=

(cid:110)

−𝑖𝐻𝑖

(cid:16)

𝑡 +

exp

1
(cid:214)

𝑖=𝐿

(cid:17)

(cid:111)

𝜏

𝜏

2

𝐿
(cid:214)

𝑖=1

(cid:110)

−𝑖𝐻𝑖

(cid:16)

𝑡 +

exp

(cid:17)

(cid:111)

𝜏

,

𝜏

2

(5.8)

where 𝐻 is the Hamiltonian generating 𝑈. That 𝑈2 is second-order can be seen from Taylor

expanding the Dyson series of 𝑈 about 𝜏 = 0 (𝐻 must be at least twice differentiable). Moreover,

𝑈2 is time-reversal symmetric in the same sense as 𝑈: 𝑈2(𝑡, 𝑡0) = 𝑈2(𝑡0, 𝑡)†. This gives the nice
property that the error series for 𝑈 (𝑡 + 𝜏, 𝑡) − 𝑈 (𝑘) (𝑡 + 𝜏, 𝑡) has only even terms, such that higher

order formulas can be reached with approximately half the number of summands. Thus, from now

on we will be interested in the MPF

𝑈2,𝑚 (𝑡, 0) =

𝑚
∑︁

𝑗=1

𝑎 𝑗𝑈 (𝑘 𝑗 )
2

(𝑡, 0)

(5.9)

for the rest of this chapter.

We finally turn to the question of whether the time dependent MPFs of Definition 2 may be

constructed for improved approximants. At the beginning of this section, we mentioned the difficulty

presented by time-ordering in adopting the techniques from [18]. The reader of the previous chapter

may recognize that clock spaces may be used to remove time ordering, circumventing the issue.

However, when the clock variable 𝑡 is continuous, the shift term −𝐸 in the clock Hamiltonian is

108

an unbounded operator, complicating a BCH-type analysis. We conjecture, and provide a heuristic

argument, that time dependent MPFs indeed boost the approximation order for sufficiently smooth

Hamiltonians.

Conjecture 1. Let 𝐻 = (cid:205)𝐿
𝑖=1

𝐻𝑖 (𝑡), and let

𝑈2(𝑡 + 𝜏, 𝑡) =

𝑒−𝑖𝐻𝑖 (𝑡+𝜏/2)𝜏

1
(cid:214)

𝑖=𝐿

𝐿
(cid:214)

𝑖=1

𝑒−𝑖𝐻𝑖 (𝑡+𝜏/2)𝜏

be the symmetric, 2nd order Trotterized midpoint formula. Suppose each 𝐻𝑖 is 2𝑚 + 1 time

differentiable. Then the time dependent multiproduct formula 𝑈2,𝑚 (𝑡 + 𝜏, 𝑡) with base formula 𝑈2

approximates 𝑈 (𝑡 + 𝜏, 𝑡) to order 2𝑚 in 𝑡.

We now discuss a potential path to proof of this conjecture. Without loss of generality, we take

𝑡 = 0. Let 𝑘 ∈ Z+, and consider a sequence of discrete clock constructions on interval [0, 𝜏], with

parameters (𝑁 𝑝 (ℓ), 𝑁𝑞 (ℓ), 𝜎(ℓ)), such that 𝑘 always divides 𝑁𝑐 = 𝑁 𝑝 𝑁𝑞, and such that the limit

reproduces the dynamics of 𝐻 (𝑡) on the main register, as per Theorem 4.3.6. Consider one of the

elements of this sequence. Using the form of 𝐻 given in the conjecture statement, we may write

𝐶 (𝐻) =

𝐿
∑︁

𝑖=1

𝐶 (𝐻𝑖).

Thus, the clock Hamiltonian 𝐻𝑐 admits the following 2nd order symmetric Trotterization.

𝑉2(𝜏) = 𝑒𝑖Δ𝜏/2

(cid:32) 1
(cid:214)

𝑖=𝐿

𝑒−𝑖𝐶 (𝐻𝑖)𝜏/2

𝐿
(cid:214)

𝑖=1

𝑒−𝑖𝐶 (𝐻𝑖)𝜏/2

(cid:33)

𝑒𝑖Δ𝜏/2

From [18], we have that

𝑉 (𝜏) − 𝑉 𝑘

2 (𝜏/𝑘) =

𝑚−1
∑︁

𝑗=1

E2 𝑗+1(𝜏)

𝜏2 𝑗+1
𝑘 2 𝑗 + E (𝜏, 𝑘)

(5.10)

(5.11)

(5.12)

where E ∈ 𝑂 (𝜏2𝑚+1) is analytic in 𝜏. Thus the standard, well-conditioned multiproduct formula

𝑉2,𝑚 of Theorem 5.1.1 with base formula 𝑉2 satisfies

𝑉 (𝜏) − 𝑉2,𝑚 (𝜏) =

𝑚
∑︁

𝑗=1

𝑎 𝑗 E (𝜏, 𝑘 𝑗 ).

(5.13)

109

We now wish to look at the action on the main register. Applying equation (5.13) to the state

|𝜓⟩ |𝜙0⟩ of the full register, where |𝜓⟩ is arbitrary, and then taking the trace Tr𝑐 over the clock

register, one obtains

Tr𝑐 (𝑉 (𝜏) |𝜓⟩ |𝜙0⟩) − Tr𝑐 (𝑉2,𝑚 (𝜏) |𝜓⟩ |𝜙0⟩) =

𝑚
∑︁

𝑗=1

𝑎 𝑗 𝐸 (𝜏, 𝑘 𝑗 ) (|𝜓⟩)

(5.14)

where 𝐸 (𝜏, 𝑘) is a linear map on the main register defined by

𝐸 (𝜏, 𝑘)(|𝜓⟩) := Tr𝑐 (E (𝜏, 𝑘) |𝜓⟩ |𝜙0⟩).

(5.15)

The above holds for every clock space in the sequence defined by (𝑁 𝑝 (ℓ), 𝑁𝑞 (ℓ), 𝜎(ℓ)). Taking

the limit as ℓ → ∞ of equation (5.14) we may pass the limits through the finite sums and scalar

multiplications

lim
ℓ→∞

Tr𝑐 (𝑉 (𝜏) |𝜓⟩ |𝜙0⟩) − lim
ℓ→∞

Tr𝑐 (𝑉2,𝑚 (𝜏) |𝜓⟩ |𝜙0⟩) =

𝑚
∑︁

𝑗=1

𝑎 𝑗 lim
ℓ→∞

𝐸 (𝜏, 𝑘 𝑗 ) |𝜓⟩

(5.16)

provided that these limits exist. Indeed, by Theorem 4.3.6,

Tr𝑐 (𝑉 (𝜏) |𝜓⟩ |𝜙0⟩) = 𝑈 (𝜏, 0) |𝜓⟩ .

As for the MPF, taking 𝑘 steps of the Trotterization, we should find that

Tr𝑐 (𝑉2(𝜏/𝑘) 𝑘 𝜓 |𝜙0⟩𝑐) = 𝑈 (𝑘)

2

(𝜏, 0) |𝜓⟩

lim
ℓ→∞

(5.17)

(5.18)

though this must be shown. This shouldn’t be too hard, as the idea is clear: perform a sequence of

clock shifts followed by 2nd order Trotter on the main register. By passing the limit through the

multiproduct sum,

Tr𝑐 (𝑉2,𝑚 (𝜏, 0) |𝜓⟩ |𝜙0⟩𝑐) = 𝑈2,𝑚 (𝜏, 0) |𝜓⟩ .

lim
ℓ→∞

(5.19)

It remains to show that the limit limℓ 𝐸 (𝜏, 𝑘) exists, and moreover is in 𝑂 (𝜏2𝑚+1). This is

where the main challenge lies. To show that the limit of a sequence with terms of order 𝑂 (𝜏2𝑚+1)

is also 𝑂 (𝜏2𝑚+1), we can show that the 2𝑚 + 1 derivative is bounded at 𝜏 = 0. Unfortunately, our

current clock constructions have the width 𝜎 of the clock state shrinking to infinity, which means

110

the derivatives grow as well. If a different clock construction can be provided where the clock state

can have width 𝜎 ∈ 𝑂 (1), a bound can be placed and thus the limit will be 𝑂 (𝜏2𝑚+1).

Current ongoing work is being undertaken to fill in the gaps of the previous argument. How-

ever, the numerics of Section 5.7 strongly suggest that the time dependent MPFs indeed work as

expected. Moreover, the form of the time-dependent MPF of Definition 2 can be obtained by a

naive Trotterization of the continuous clock space, which is very suggestive that, beyond formal

issues, the approach is reasonable. Thus, we proceed with the assumption Conjecture 1 is true.

5.3 Time Dependent Multiproduct Simulation

Having argued that good time dependent MPFs exist, we now propose an algorithm for Hamil-

tonian simulation using these formulas. We will provide some accompanying discussion to explain

our choices, and at the end we will more directly state the approach.

In order to present a concrete computational model for our Hamiltonian, we further specify that

our LCH Hamiltonian 𝐻 (𝑡) is of the form

𝐻 (𝑡) =

𝐿
∑︁

𝑖=1

𝛼𝑖 (𝑡)𝐻𝑖

(5.20)

where each 𝛼𝑖 (𝑡) ∈ R is assumed 2𝑚+1 differentiable for an 𝑚-term MPF. Without loss of generality

we take ∥𝐻𝑖 ∥ ≤ 1.

From the onset, there are a couple of choices to make. The MPFs, in principle, could approxi-

mate the entire interval [0, 𝑇] provided that the Trotter steps 𝑘𝑖 are sufficiently large. However, this

has several disadvantages. First, there is no flexibility to treat some subintervals of [0, 𝑇] as more

difficult than others and allocate resources appropriately. Second, the well-conditioned scheme

of [90] would have to be abandoned or modified to accommodate larger (cid:174)𝑘.

Instead, we divide

[0, 𝑇] into a mesh of 𝑟 subintervals, not necessarily uniform, but rather constructed to account

for more difficult parts of the simulation. We provide a greedy algorithm for constructing such a

mesh in Section 5.9. The algorithm requires a computable Λ2𝑚+1-bound to work (see Definition 3),

however, a practitioner might prefer a more heuristic approach to constructing the time mesh. For

the moment, we will simply say that, given 𝑡𝑖, the next time point 𝑡𝑖+1 is incremented roughly as

111

1/Λ2𝑚+1(𝑡) for 𝑡 in a neighborhood of 𝑡𝑖, where Λ2𝑚+1 is a positive real-valued function of 𝐻 and

its derivatives that grows for larger or faster fluctuating 𝐻.

Once the mesh points 𝑡0, 𝑡1, . . . , 𝑡𝑟 are determined, a time dependent MPF is performed over each

subinterval [𝑡𝑖, 𝑡𝑖+1] in sequence. We assume the MPF is implemented using the LCU technique.

The base midpoint formula 𝑈2 must be implemented by some scheme which depends on the

structure of 𝐻 (𝑡), though the approximating unitary 𝑊2 should be at least 2nd-order and preserve

the time-reversal symmetry of 𝑈2 (and 𝑈). It is known that product formulas exhibit commutator

scaling, meaning that, in the limit where all 𝐻 𝑗 commute pairwise and all 𝛼 𝑗 are constant functions,

the simulation error goes to zero. Hence, the MPF will also inherit this desirable property. It is for

precisely this reason that, in Table 5.1, we claim our MPFs exhibit commutator scaling.

Let us now supply our procedure for the multiproduct simulation. Given fundamental parame-

ters, [0, 𝑇], 𝜖, and a description of 𝐻 (𝑡):

1. Compute a Λ2𝑚+1 bound (Definition 3) for some 𝑚 larger than the expected number of MPF

terms. This is more a less a bound on the generalized "size" of 𝐻 (𝑡).

2. Construct a time mesh of 𝑟 steps using the algorithm of Section 5.9.

3. Perform a sequence of MPFs over each time slice, with 2nd order base formula 𝑊2 approxi-

mating the midpoint formula.

Not much more about the parameter choices, such as 𝑚 or 𝑟, can be said without an error analysis.

This will be supplied in the following section. We then return to the question of algorithmic cost

via a query model.

5.4 Error Analysis

In this section, we analyze the errors arising between the exact unitary 𝑈 and the MPF approx-

imation ˜𝑈 given by

𝑟
(cid:214)

˜𝑈 (𝑇, 0) =

𝑈2,𝑚 (𝑡𝑖, 𝑡𝑖−1).

(5.21)

𝑖=1
This analysis will ignore hardware imperfections and decoherence, assume that 𝑈2 is implemented

perfectly, and assume exact coefficients 𝑎 𝑗 . In the query complexity analysis of Section 5.6 we will

112

consider additional algorithmic errors arising from a more precise specification of the Hamiltonian

input model.

We introduce a useful definition to quantify errors succinctly. It is well understood that MPFs,

like regular product formulas, have smoothness requirements to ensure convergence. To quantify

errors and costs of MPFs, we provide a metric which captures the "size" of 𝐻 and its derivatives at

each point in time, in order to characterize the difficulty of simulation.

Definition 3. Let 𝐻 (𝑡) = (cid:205)𝐿
𝑖=1

𝛼𝑖 (𝑡)𝐻𝑖 be a time dependent, finite-dimensional Hamiltonian with

𝐻𝑖 Hermitian and 𝛼𝑖 (𝑡) ∈ R having 𝑛 ∈ N ∪ {∞} continuous derivatives. For each 𝑖 define a

Λ𝑖,𝑛-bound ("Lambda i n bound") as any continuous function Λ𝑖,𝑛 : [0, 𝑇] → R+ satisfying the

following bounds with respect to 𝐻 and its derivatives

Λ𝑖,𝑛 (𝑡) ≥ sup
𝑗 ∈[𝑛]

𝑗+1√︃

∥𝛼( 𝑗)
𝑖

(𝑡) ∥ ∀𝑡 ∈ [0, 𝑇]

where 𝑓 (𝑛) represents an 𝑛th derivative of 𝑓 , and [𝑛] := { 𝑗 ∈ N | 𝑗 ≤ 𝑛}. Assuming such bounds

exist for all 𝑖 = 1, . . . , 𝐿, we say that 𝐻 (𝑡) is Λ𝑛-bounded. We further say that 𝐻 (𝑡) is Λ𝑛-boundable

if it admits some Λ𝑛-bound. For convenience, we define Λ𝑖 ≡ Λ𝑖,∞. We also define a Λ𝑛 bound as

any continuous on [0, 𝑇] satisfying

For near-constant 𝛼𝑖 (𝑡), Λ𝑖,𝑛 is simply an upper bound on |𝛼𝑖 |, while for rapid oscillations

Λ𝑛 (𝑡) ≥ max
𝑖∈[𝐿]

Λ𝑖,𝑛 (𝑡).

the derivative terms will dominate. Observe that for finite 𝑛, our assumptions imply that Λ𝑖,𝑛 (𝑡)
exists (𝐻 is Λ𝑖,𝑛-boundable), since |𝛼( 𝑗)

| is continuous on a compact interval and hence a bounded

𝑖

function. Also in the finite case, the supremum may be replaced with a simple max, and Λ𝑖,𝑛 (𝑡)

may be taken as equal to the right hand side because it is the maximum of a finite set of continuous

functions, which is continuous. For this "minimal choice," Λ𝑖,𝑛 (𝑡) is a nondecreasing sequence in 𝑛.

For each 𝑛, there also exists a Λ𝑖,𝑛 that is constant in 𝑡. Allowing Λ𝑖,𝑛 to vary in time, however, takes

into consideration the possibility that the expense of simulating 𝐻 will vary with time. We note

that Λ𝑖,𝑛-bounds are additive in the sense that, for 𝐻 (𝑡) and 𝐺 (𝑡) admitting Λ𝐻

𝑖,𝑛 and Λ𝐺

𝑖,𝑛-bounds,

respectively, Λ𝐻

𝑖,𝑛 + Λ𝐺

𝑖,𝑛 is a Λ𝑖,𝑛-bound on 𝐻 + 𝐺.

113

In contrast to finite 𝑛, the existence of a Λ𝑖,∞-bound is not guaranteed, and amounts to the

assumption that the derivatives of 𝐻 grow at most exponentially for asymptotically large 𝑗 and

fixed 𝑡. There are smooth, even analytic functions which do not satisfy this, many of which are

physically interesting. A simple example is a Gaussian pulse

𝛼(𝑡) = 𝑒−𝑡2

(5.22)

whose derivatives, generating the Hermite polynomials, grow factorially with 𝑛 at 𝑡 = 0. Other in-

teresting cases, such as harmonic oscillations or exponential growth and decay, do admit a Λ-bound.

Despite these restrictions, we adopt this approach for simplicity and in order to facilitate compari-

son with prior work on general-order Suzuki-Trotter formulas [137]. Admittedly, a modification of

Definition 3 to be an upper bound on

𝑗 −1 𝑗+1√︃

∥𝛼( 𝑗)
𝑖

(𝑡) ∥

max
𝑗

(5.23)

would expand the class of functions admitting Λ-bounds to analytic functions (though not generic

smooth functions).

We now begin the error analysis of (5.21) in earnest. From a triangle inequality the error can

be bounded as the error in each step.

∥𝑈 (𝑇, 0) − ˜𝑈 (𝑇, 0) ∥ ≤

𝑟
∑︁

𝑖=1

∥𝑈 (𝑡𝑖, 𝑡𝑖−1) − 𝑈2,𝑚 (𝑡𝑖, 𝑡𝑖−1) ∥

(5.24)

Therefore, to ensure an error at most 𝜖, it suffices that each subinterval has error at most 𝜖/𝑟. We

thus focus a single subinterval. An upper bound on this error is supplied by the following theorem,

which the main technical result of this section.

Theorem 5.4.1. Let 𝐻 : [𝑡0, 𝑡1] → Herm(H ) be a time dependent Hamiltonian on finite-

dimensional H with 2𝑚 + 1 continuous derivatives on [𝑡0, 𝑡1] and Λ2𝑚+1-bound. Suppose further

that

𝑒𝐿 max
𝜏∈[𝑡0,𝑡1]

Λ2𝑚+1(𝜏) (𝑡1 − 𝑡0) < 1.

Then for any 𝑚 ∈ Z+ there exists (cid:174)𝑘 ∈ Z𝑚

∥𝑈 (𝑡1, 𝑡0) − 𝑈2,𝑚 (𝑡1, 𝑡0)∥ <

+ and 𝑎 ∈ R𝑚 such that
(cid:18)

∥𝑎∥1√
𝜋𝑚

5𝐿 max
𝜏∈[𝑡0,𝑡1]

Λ2𝑚+1(𝜏) (𝑡1 − 𝑡0)

(cid:19) 2𝑚+1

114

and ∥𝑎∥1 ∈ 𝑂 (log(𝑚)).

Observe that convergence of the above error bound to zero as 𝑚 → ∞ is conditioned on

sufficiently small 𝑡1 − 𝑡0. This is potentially unsurprising, as the Suzuki-Trotter formulas also do not

provide an unconditionally converging sequence of approximations to the time evolution operator.

Note as well the parallel roles between 𝑚 and the Suzuki-Trotter order 𝑘 in reducing the error. In

our case, however, we shall see that the simulation cost increases only polynomially in 𝑚, whereas

for product formulas the cost is necessarily exponential in 𝑘.

The term ∥𝑎∥1/

√

𝜋𝑚 is 𝑜(1) for large 𝑚 and can be more or less ignored. Unfortunately, the

Λ2𝑚+1 scales as the worst 𝛼𝑖 times the number of terms 𝐿, which seems too cynical. However,

improving on this may greatly complicate the proof of the error bound. Theorem 5.4.1 will be

the important result that informs the algorithmic choices and complexity analysis of subsequent

sections.

Having characterized the error on a single subinterval of [0, 𝑇], the full error over 𝑟 subintervals

may be found simply using (5.24).

We prove Theorem 5.4.1 using a similar strategy to that used to provide error estimates for the

Suzuki-Trotter formulas [14, 137, 26]. As 𝐻 is continuously differentiable at least 2𝑚 + 1 times,

𝑈2,𝑚 is a valid extrapolant under our , and cancels the first 𝑚 terms in the error series. We can thus

express the difference 𝑈2,𝑚 − 𝑈 using the integral Taylor remainder formulas

with

𝑈2,𝑚 (𝑡, 𝑡0) − 𝑈 (𝑡, 𝑡0) = 𝑅2𝑚 − R2𝑚

R2𝑚 :=

𝑅2𝑚 :=

1
2𝑚!
1
2𝑚!

∫ 𝑡

𝑡0
∫ 𝑡

𝑡0

(𝑡 − 𝜏)2𝑚𝑈 (2𝑚+1) (𝜏, 𝑡0)𝑑𝜏

(𝑡 − 𝜏)2𝑚𝑈 (2𝑚+1)

2,𝑚

(𝜏, 𝑡0)𝑑𝜏,

where 𝑈 (𝑛) refers to derivatives in the first argument. By the triangle inequality,

∥𝑈2,𝑚 (𝑡, 𝑡0) − 𝑈 (𝑡, 𝑡0) ∥ ≤ ∥R2𝑚 ∥ + ∥𝑅2𝑚 ∥

115

(5.25)

(5.26)

(5.27)

(5.28)

and we upper bound each remainder in separate lemmas.

The easier bound is R2𝑚, so we begin with the corresponding lemma.

Lemma 5.4.2. The remainder term R2𝑚 in equation (5.27) satisfies

∥R2𝑚 ∥ <

(cid:18)

1
√
𝜋𝑚

2

2𝐿 max
𝜏∈[𝑡0,𝑡]

Λ2𝑚+1(𝜏) (𝑡 − 𝑡0)

(cid:19) 2𝑚+1

.

Proof. Recall that 𝑈, as the exact propagator, satisfies the Schrödinger equation (2.16). Higher

derivatives can easily be found through repeated application of the product rule. The result will

be a polynomial in the derivatives of 𝐻 times 𝑈 itself. Under the spectral norm, using the triangle

and submultiplicative properties, the ordering of terms doesn’t matter, and therefore equivalent

to the expression one gets taking derivatives of a scalar exponential. Noting that ∥𝑈 ∥ = 1, the

resulting polynomial is the complete exponential Bell polynomial from Faà di Bruno’s formula (see

Section 2.8). Letting 𝑛 = 2𝑚 + 1, we have

∥𝜕𝑛

𝑡 𝑈 (𝑡, 𝑡0)∥ ≤ 𝑌𝑛

(cid:16)

∥𝐻 (𝑡) ∥, ∥ (cid:164)𝐻 (𝑡) ∥, . . . , ∥𝐻 (𝑛−1) (𝑡) ∥

(cid:17)

.

(5.29)

From the definition of Λ𝑖,𝑛, we have

∥𝐻 ( 𝑗) (𝑡) ∥ ≤

≤

𝐿
∑︁

𝑖=1
∑︁

𝑖

|𝛼( 𝑗)
𝑖

(𝑡)|

Λ𝑖,𝑛 (𝑡) 𝑗+1

≤ (𝐿Λ𝑛 (𝑡)) 𝑗+1

and since the Bell polynomials 𝑌𝑛 are monotonic in each argument,

(cid:16)

𝑌𝑛

∥𝐻 ∥, ∥ (cid:164)𝐻 ∥, . . . , ∥𝐻 (𝑛−1) ∥

(cid:17)

≤ 𝑌𝑛 (𝐿Λ𝑛 (𝑡), (𝐿Λ𝑛 (𝑡))2, . . . , (𝐿Λ𝑛 (𝑡))𝑛)

where 𝑏𝑛 are the Bell numbers (Section 2.8). Thus,

= (𝐿Λ𝑛 (𝑡))𝑛𝑏𝑛

∥𝜕𝑛

𝑡 𝑈 (𝑡, 𝑡0) ∥ ≤ (𝐿Λ𝑛 (𝑡))𝑛𝑏𝑛.

116

(5.30)

(5.31)

(5.32)

Finally, returning to the bound on R2𝑚, we have from the integral triangle inequality that

∥R2𝑚 ∥ ≤

≤

1
(2𝑚)!
1
(2𝑚)!

∫ 𝑡

𝑡0
∫ 𝑡

𝑡0

(𝑡 − 𝜏)2𝑚 ∥𝜕2𝑚+1

𝜏

𝑈 (𝜏, 𝑡0) ∥𝑑𝜏

(𝑡 − 𝜏)2𝑚 (𝐿Λ2𝑚+1(𝜏))2𝑚+1𝑏2𝑚+1𝑑𝜏

(5.33)

where we made use of equation (5.32). This, in turn, can be bounded by maximizing Λ2𝑚+1 over

[𝑡0, 𝑡].

∥R2𝑚 ∥ ≤

𝑏2𝑚+1
(2𝑚)!

Λ2𝑚+1(𝜏))2𝑚+1

∫ 𝑡

𝑡0

𝑑𝜏(𝑡 − 𝜏)2𝑚

(𝐿 max
𝜏∈[𝑡0,𝑡]
(cid:18)

≤

𝑏2𝑚+1
(2𝑚 + 1)!

𝐿 max
𝜏∈[𝑡0,𝑡]

Λ2𝑚+1(𝜏) (𝑡 − 𝑡0)

(cid:19) 2𝑚+1

(5.34)

Finally, we upper bound the prefactor using a Stirling bound and bounds from [13] on the Bell

numbers. For all 𝑚 ∈ Z+,

𝑏2𝑚+1
(2𝑚 + 1)!

<

(cid:17) 2𝑚+1

(cid:16) 0.792(2𝑚+1)
log(2𝑚+2)
√︁2𝜋(2𝑚 + 1)

(cid:17) 2𝑚+1

(cid:16) 2𝑚+1
𝑒

=

1
√︁2𝜋(2𝑚 + 1)

(cid:18)

.792𝑒
log(2𝑚 + 2)

(cid:19) 2𝑚+1

.

Plugging this into equation (5.34),

∥R2𝑚 ∥ <

<

1
√︁2𝜋(2𝑚 + 1)
(cid:18)

1
√
𝜋𝑚
2

2𝐿 max
𝜏∈[𝑡0,𝑡]

(cid:18)

0.792𝑒
log(2𝑚 + 2)

𝐿 max
𝜏∈[𝑡0,𝑡]

Λ2𝑚+1(𝜏) (𝑡 − 𝑡0)

(cid:19) 2𝑚+1

Λ2𝑚+1(𝜏) (𝑡 − 𝑡0)

(cid:19) 2𝑚+1

.

The last line is the result of the lemma.

We now state the bound on the Taylor 𝑅2𝑚 for the time dependent MPF.

Lemma 5.4.3. In the notation above, suppose that

𝑒𝐿 max
𝜏∈[𝑡0,𝑡1]

Λ2𝑚+1(𝜏) (𝑡1 − 𝑡0) < 1.

Then the remainder term 𝑅2𝑚 in equation (5.27) satisfies

∥𝑅2𝑚 ∥ <

(cid:18)

∥𝑎∥1
√
𝜋𝑚
2

5𝐿 max
𝜏∈[𝑡0,𝑡]

Λ2𝑚+1(𝜏) (𝑡 − 𝑡0)

(cid:19) 2𝑚+1

.

117

(5.35)

(5.36)

□

The proof is more technical than the previous bound, and is given at the end of this section.

First, we quickly prove Theorem 5.4.1 assuming the truth of the above Taylor remainder lemmas.

Proof of Theorem 5.4.1. First, we note that ∥𝑎∥1 ≥ 1, since 𝑎 necessarily satisfies (cid:205) 𝑗 𝑎 𝑗 = 1 from

the Vandermonde constraints (5.2). From equation (5.28), the error ∥𝑈 (𝑡, 𝑡0) − 𝑈2,𝑚 (𝑡, 𝑡0)∥ is

bounded by the sum of the remainder upper bounds derived in Lemmas 5.4.3 and 5.4.2. Comparing

the two, we see that 𝑅2𝑚 dominates R2𝑚 for all 𝑚 ≥ 1. We thus take twice the larger as an upper

bound

∥𝑈 (𝑡, 𝑡0) − 𝑈2,𝑚 (𝑡, 𝑡0)∥ < 2∥𝑅2𝑚 ∥
(cid:18)

<

∥𝑎∥1√
𝜋𝑚

5𝐿 max
𝜏∈[𝑡0,𝑡]

This completes the proof.

Λ2𝑚+1(𝜏) (𝑡 − 𝑡0)

(cid:19) 2𝑚+1

.

(5.37)

□

To prove Lemma 5.4.3, we will first need a technical lemma that bounds the size of ordinary

exponentials of time dependent matrices.

Lemma 5.4.4. Let 𝐴(𝑡) be an anti-Hermitian valued function of 𝑡 ∈ R with 𝑛 bounded derivatives.

Then

∥𝑑𝑛

𝑡 𝑒 𝐴(𝑡) ∥ ≤ 𝑌𝑛

(cid:16)

∥𝑑𝑡 𝐴(𝑡) ∥, ∥𝑑2

𝑡 𝐴(𝑡) ∥, . . . , ∥𝑑𝑛

𝑡 𝐴(𝑡) ∥

(cid:17)

where 𝑌𝑛 is the complete exponential Bell polynomial.

In the scalar case, Faà di Bruno’s bound is an exact expression (see Section 2.8), so the content

of our result is that a corresponding bound holds even in the non-scalar case. The exponential

disappears because 𝑒 𝐴(𝑡) is unitary.

Proof of Lemma 5.4.4. From the Trotter product theorem, we have

𝑡 exp( 𝐴(𝑡)) = 𝜕𝑛
𝜕𝑛
𝑡

(exp( 𝐴(𝑡)/𝑟))𝑟 .

lim
𝑟→∞

(5.38)

Using the fact that the series converges uniformly, we may interchange the order of differentiation

and the limit. This leads to

∥𝜕𝑛

𝑡 exp( 𝐴(𝑡))∥ ≤ lim
𝑟→∞

(cid:18)

∑︁

𝑆

𝑛
𝑠1, . . . , 𝑠𝑟

(cid:19)

𝑟
(cid:214)

𝑞=1

(cid:13)𝜕𝑠𝑞
(cid:13)

𝑡 exp( 𝐴(𝑡)/𝑟)(cid:13)
(cid:13) .

(5.39)

118

Here the sum over 𝑆 is constrained such that 𝑠 𝑗 ≥ 0 and 𝑠1 + · · · + 𝑠𝑟 = 𝑛. Then using Taylor’s

theorem we have

(cid:13)𝜕𝑠𝑞
(cid:13)

𝑡 exp( 𝐴(𝑡)/𝑟)(cid:13)

(cid:13) ≤

∥ 𝐴(𝑠𝑞) (𝑡) ∥
𝑟

+ 𝑂 (1/𝑟 2).

(5.40)

for 𝑠𝑞 > 0, where the 𝑂 (1/𝑟 2) terms will vanish as 𝑟 → ∞. The 𝑠𝑞 = 0 case has upper bound 1 by

unitarity. Hence, put together,

∥𝜕𝑛

𝑡 exp( 𝐴(𝑡))∥ ≤ lim
𝑟→∞

(cid:18)

∑︁

𝑛
𝑠1, . . . , 𝑠𝑟

(cid:19)

𝑟
(cid:214)

(cid:32) ∥ 𝐴(𝑠𝑞) (𝑡) ∥(1 − 𝛿𝑠𝑞,0)
𝑟

(cid:33)

+ 𝛿𝑠𝑞,0

.

(5.41)

𝑞=1
Now let us define a scalar function 𝑎(𝑥) defined for 𝑥 in a neighborhood of 𝑡 such that, for any

𝑆

𝑘 such that 0 ≤ 𝑘 ≤ 𝑛,

𝑎 (𝑘) (𝑡) = ∥ 𝐴(𝑘) (𝑡) ∥(1 − 𝛿𝑘,0)

(5.42)

for a particular 𝑥 = 𝑡. Such a function can be seen to exist by considering the 𝑛th degree Taylor

polynomial. We may apply the standard Faà di Bruno formula (2.49) to 𝑎, so that

𝑥 𝑒𝑎(𝑥)
𝜕𝑛

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝑡

= 𝑒𝑎(𝑡)𝑌𝑛 (∥ 𝐴(1) (𝑡)∥, . . . , ∥ 𝐴(𝑛) (𝑡) ∥) = 𝑌𝑛 (∥ 𝐴(1) (𝑡) ∥, . . . , ∥ 𝐴(𝑛) (𝑡) ∥).

(5.43)

On the other hand we can split 𝑎(𝑡) into 𝑟 steps and compute the 𝑛th derivative, just as for the

Trotter product theorem.

𝑥 𝑒𝑎(𝑥)
𝜕𝑛

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝑡

= lim
𝑟→∞

(cid:18)

∑︁

𝑆

𝑛
𝑠1, . . . , 𝑠𝑟

(cid:19)

𝑟
(cid:214)

𝑞=1

(cid:32) ∥ 𝐴(𝑠𝑞) (Δ𝑡) ∥(1 − 𝛿𝑠𝑞,0)
𝑟

(cid:33)

+ 𝛿𝑠𝑞,0

By comparing expressions (5.41) and (5.44), we see that

∥𝜕𝑛

𝑡 exp 𝐴(𝑡) ∥ ≤ 𝜕𝑛

𝑥 𝑒𝑎(𝑥)

(cid:12)
(cid:12)
(cid:12)
(cid:12)𝑥=𝑡

and applying (5.43), we reach our desired bound Faà di Bruno bound.

∥𝜕𝑛

𝑡 exp( 𝐴(𝑡))∥ ≤ 𝑌𝑛 (∥ 𝐴(1) (𝑡) ∥, . . . , ∥ 𝐴(𝑛) (𝑡) ∥)

(5.44)

(5.45)

(5.46)

We evaluate the derivatives of 𝐴(𝑡), and express them in terms of the derivatives of the

Hamiltonian, 𝐻 ( 𝑗) (for simplicity, we leave off the evaluation point. The derivative is with respect

to the Hamiltonian’s single argument). The result is

𝜕 𝑗
𝑡 𝐴(𝑡) =

−𝑖
𝑘

(cid:34)(cid:18) 𝑞 − 1/2
𝑘

(cid:19) 𝑗

(𝑡 − 𝑡0)𝐻 ( 𝑗) + 𝑗

(cid:18) 𝑞 − 1/2
𝑘

(cid:19) 𝑗−1

(cid:35)

.

𝐻 ( 𝑗−1)

(5.47)

119

Employing the Λ𝑛-bound from Definition 3, we have that

∥𝜕 𝑗

𝑡 𝐴(𝑡)∥ ≤

(cid:34) (cid:18) 𝑞 − 1/2
𝑘

1
𝑘

(cid:19) 𝑗

(𝑡 − 𝑡0)Λ

𝑗+1
𝑛,𝑞 + 𝑗

(cid:19) 𝑗−1

(cid:18) 𝑞 − 1/2
𝑘

(cid:35)

Λ𝑛,𝑞𝑞 𝑗

(cid:19) 𝑗

(cid:18) 𝑞 − 1/2
𝑘

=

𝑗
𝑛,𝑞

Λ

(cid:20)

𝑗
𝑞 − 1/2

+

1
𝑘

(𝑡 − 𝑡0)Λ𝑛,𝑞

(cid:21)

.

Here,

Λ𝑛,𝑞 := max
𝜏∈𝐼𝑞

Λ𝑛 (𝜏)

(5.48)

(5.49)

and 𝐼𝑞 = [𝑡0 + (𝑞 − 1)(𝑡 − 𝑡0)/𝑘, 𝑡0 + 𝑞(𝑡 − 𝑡0)/𝑘] is the 𝑞th interval in the mesh from 𝑡0 to 𝑡 with 𝑘

even spaces. Since Λ𝑛,𝑞 ≤ max𝜏∈[𝑡0,𝑡] Λ𝑛 (𝜏), from the assumptions of the lemma, Λ𝑛,𝑞 (𝑡 − 𝑡0) < 1.

Hence,

where ˜Λ𝑛,𝑞 ≡ Λ𝑛,𝑞 (𝑞 − 1/2)/𝑘.

∥𝜕 𝑗

𝑡 𝐴(𝑡) ∥ ≤ ˜Λ

𝑗
𝑛,𝑞

(cid:20)

𝑗
𝑞 − 1/2

(cid:21)

1
𝑘

+

(5.50)

Plugging this into the formula into (5.46) and using the definition of 𝑌𝑛 given by (2.50), our

bound becomes

∥𝜕𝑛

𝑡 𝑈2(𝑡)∥ ≤

∑︁

𝐶

𝑛!
𝑐1! . . . 𝑐𝑛!

𝑛
(cid:214)

(cid:32) (

𝑗=1

𝑘 ) ˜Λ

𝑗

𝑞−1/2 + 1
𝑗!

(cid:33) 𝑐 𝑗

𝑗
𝑛,𝑞

.

Using the sum property of the coefficients 𝑐 𝑗 , we can move the ˜Λ

𝑗
𝑛,𝑞 out of the sum.

∥𝜕𝑛

𝑡 𝑈2(𝑡)∥ ≤

(cid:18)

Λ𝑛,𝑞

𝑞 − 1/2
𝑘

(cid:19) 𝑛

∑︁

𝐶

𝑛!
𝑐1! . . . 𝑐𝑛!

(cid:32)

𝑛
(cid:214)

𝑗=1

(cid:18)

=

Λ𝑛,𝑞

(cid:19) 𝑛

𝑞 − 1/2
𝑘

𝑌𝑛

(cid:17)

(cid:16)

(cid:174)𝑥 (𝑛)
𝑞,𝑘

(cid:33) 𝑐 𝑗

𝑗

𝑞−1/2 + 1
𝑗!

𝑘

(5.51)

(5.52)

In the last line, we reapplied the definition of 𝑌𝑛 and of the vectors (cid:174)𝑥 (𝑛)

𝑞,𝑘 . This completes our bound

for the 𝑈2 formula for the 𝑞th segment of mesh defined by 𝑘 𝑗 .

□

We conclude this section with a proof of the bound on 𝑅2𝑚.

Proof of Lemma 5.4.3. Without loss of generality, we take 𝑡0 = 0. The relevant expressions are

𝑈2,𝑚 (𝑡, 0) =

𝑎 𝑗𝑈 (𝑘 𝑗 )
2

(𝑡, 0)

𝑚
∑︁

𝑗=1

120

(5.53)

and

𝑈 (𝑘)
2

(𝑡, 0) :=

𝑘
(cid:214)

ℓ=1

𝑈2(𝑡ℓ, 𝑡ℓ−1)

with 𝑡ℓ := 𝑡ℓ/𝑘. The Taylor remainder in integral form is given by

𝑅2𝑚 =

=

1
(2𝑚)!
1
(2𝑚)!

∫ 𝑡

0
𝑚
∑︁

𝑗=1

(𝑡 − 𝜏)2𝑚 𝑑2𝑚+1
𝑑𝜏2𝑚+1
(𝑡 − 𝜏)2𝑚 𝑑2𝑚+1
𝑑𝜏2𝑚+1

∫ 𝑡

𝑎 𝑗

0

𝑈2,𝑚 (𝜏, 0)𝑑𝜏

𝑈 (𝑘 𝑗 )
2

(𝜏, 0)𝑑𝜏.

With a couple triangle inequalities, this is upper bounded as

∥𝑅2𝑚 ∥ ≤

≤

𝑚
∑︁

1
(2𝑚)!
∥𝑎∥1
(2𝑚 + 1)!

𝑗=1

|𝑎 𝑗 |

𝑡2𝑚+1
2𝑚 + 1

max
𝜏∈[0,𝑡]

∥𝑑2𝑚+1

𝜏 𝑈 (𝑘 𝑗 )

2

(𝜏, 0) ∥

𝑡2𝑚+1 max
𝑗,𝜏

∥𝑑2𝑚+1

𝜏 𝑈 (𝑘 𝑗 )

2

(𝜏, 0) ∥

(5.54)

(5.55)

(5.56)

where in the last line we employed a Hölder inequality. Our focus is now on bounding the derivative,

which we unravel layer by layer using frequent multinomial expansions. First,

𝜏𝑈 (𝑘)
𝑑𝑛

2

(𝜏, 0) =

(cid:18)

∑︁

𝑁

𝑛
𝑛1, . . . , 𝑛𝑘

(cid:19)

𝑘
(cid:214)

ℓ=1

𝑑𝑛ℓ
𝜏 𝑈2(𝜏ℓ, 𝜏ℓ−1).

(5.57)

Next, we write

where

𝑈2(𝜏ℓ, 𝜏ℓ−1) =

=

1
(cid:214)

𝑖=𝐿
2𝐿
(cid:214)

𝑖=1

𝑒−𝑖𝐻𝑖𝛼𝑖 (𝜏ℓ −1/2)𝜏/𝑘

𝐿
(cid:214)

𝑖=1

𝑒−𝑖𝐻𝑖𝛼𝑖 (𝜏ℓ −1/2)𝜏/𝑘

𝑒 𝐴𝑖,ℓ

𝐴𝑖,ℓ := −𝑖𝐻𝑖𝛼𝑖 (𝜏ℓ−1/2)𝜏/𝑘

and 𝑖 is defined by reflection for 𝑖 > 𝐿. Once again performing a multinomial expansion,

𝑑𝑛
𝜏𝑈2(𝜏ℓ, 𝜏ℓ−1) =

(cid:18)

∑︁

𝑁

𝑛
𝑛1, . . . , 𝑛2𝐿

(cid:19) 2𝐿
(cid:214)

𝑖=1

𝜏 𝑒 𝐴𝑖,ℓ .
𝑑𝑛𝑖

We now bound the individual ordinary operator exponentials. Invoking Lemma 5.4.4,

∥𝑑𝑛

𝜏 𝑒 𝐴𝑖,ℓ ∥ ≤ 𝑌𝑛 (cid:0)∥𝑑𝜏 𝐴𝑖,ℓ ∥, . . . , ∥𝑑𝑛

𝜏 𝐴𝑖,ℓ ∥(cid:1) .

121

(5.58)

(5.59)

(5.60)

(5.61)

In turn, we have

𝑑𝑛
𝜏 𝐴𝑖,ℓ = −𝑖

= −𝑖

𝐻𝑖
𝑘
𝐻𝑖
𝑘

𝑑𝑛
𝜏 (𝛼𝑖 (𝜏ℓ−1/2)𝜏)
(cid:34) (cid:18) ℓ − 1/2
𝑘

(cid:19) 𝑛

𝜏𝛼(𝑛)
𝑖

(𝜏ℓ−1/2) + 𝑛

(cid:19) 𝑛−1

(cid:18) ℓ − 1/2
𝑘

𝛼(𝑛−1)
𝑖

(𝜏ℓ−1/2)

(cid:35)

(5.62)

where 𝛼(𝑛) (𝑥) refers to the 𝑛th derivative of 𝛼 with respect to its argument, then evaluated at 𝑥 (i.e.,

not a 𝜏 derivative). Since ∥𝐻𝑖 ∥ ≤ 1 we have

∥𝑑𝑛

𝜏 𝐴𝑖,ℓ ∥ < 1
𝑘

(ℓ/𝑘)𝑛−1 (cid:16)

(ℓ/𝑘)𝜏|𝛼(𝑛)

𝑖

(𝜏ℓ−1/2)| + 𝑛|𝛼(𝑛−1)

𝑖

(𝜏ℓ−1/2)|

(cid:17)

.

(5.63)

From Definition 3, 𝛼( 𝑗)

𝑖

(𝑡) ≤ Λ𝑖,𝑛 (𝑡) 𝑗+1. Dropping the 𝑛 and 𝑡 dependence for the moment,

∥𝑑𝑛

𝜏 𝐴𝑖,ℓ ∥ < (ℓ/𝑘)𝑛 (cid:16)

(𝜏/𝑘)Λ𝑛+1

𝑖

+ (𝑛/ℓ)Λ𝑛
𝑖

(cid:17)

= (Λ𝑖ℓ/𝑘)𝑛 (Λ𝑖𝜏/𝑘 + 𝑛/ℓ) .

(5.64)

We’ve reached the bottom, and now proceed to work our way back up to the Taylor remainder 𝑅2𝑚,

starting with (5.61). Using Lemma 5.4.4,

(cid:32)

𝑛
(cid:214)

(cid:33) 𝑐 𝑗

∥𝑑 𝑗

𝜏 𝐴𝑖,ℓ ∥
𝑗!

∥𝑑𝑛

𝜏 𝑒 𝐴𝑖,ℓ ∥ ≤

∑︁

𝐶

𝑛!
𝑐1!𝑐2! . . . 𝑐𝑛!

< ∑︁
𝐶

𝑛!
𝑐1!𝑐2! . . . 𝑐𝑛!

𝑗=1
𝑛
(cid:214)

𝑗=1

(cid:18) (Λ𝑖ℓ/𝑘) 𝑗 (Λ𝑖𝜏/𝑘 + 𝑗/ℓ)
𝑗!

(cid:19) 𝑐 𝑗

.

(5.65)

Using the sum rule for 𝐶 we can pull out a factor of (Λ𝑖ℓ/𝑘). With the upper bound 𝑗 ≤ 𝑛 and the

monotonicity of 𝑌𝑛, we obtain the bound

∥𝑑𝑛

𝜏 𝑒 𝐴𝑖,ℓ ∥ < (Λ𝑖ℓ/𝑘)𝑛𝐵𝑛 (Λ𝑖𝜏/𝑘 + 𝑛/ℓ)

where 𝐵𝑛 is the Bell polynomial (see Section 2.8). For simplicity, define

𝑥𝑖,ℓ,𝑛 = Λ𝑖𝜏/𝑘 + 𝑛/ℓ

as the argument to 𝐵𝑛. Employing the bound (2.56),

∥𝑑𝑛

𝜏 𝑒 𝐴𝑖,ℓ ∥ < (Λ𝑖ℓ/𝑘)𝑛

(cid:18)

𝑛
log(1 + 𝑛/𝑥𝑖,ℓ,𝑛)

(cid:19) 𝑛

122

(5.66)

(5.67)

(5.68)

which is valid for all 𝑛 > 0, and for 𝑛 = 0 when defined by the 0+ limit. We can simplify the

reciprocal log with the bound

1
log(1 + 𝑛/𝑥𝑖,ℓ,𝑛)

<

=

(cid:18) 1
2
1
2𝑛

+

(cid:18)

(cid:19) 𝑛

𝑥𝑖,ℓ,𝑛
𝑛
2𝑥𝑖,ℓ,𝑛
𝑛

1 +

(cid:19) 𝑛

.

This gives us the simplified exponential derivative

∥𝑑𝑛

𝜏 𝑒 𝐴𝑖,ℓ ∥ < (Λ𝑖ℓ/2𝑘)𝑛 (𝑛 + 2𝑥𝑖,ℓ,𝑛)𝑛.

We now move up a level to reconsider (5.60). Employing a triangle inequality,

∥𝑑𝑛

𝜏𝑈2(𝜏ℓ, 𝜏ℓ−1)∥ ≤

∑︁

𝑁

< ∑︁
𝑁

(cid:18)

(cid:18)

𝑛
𝑛1, . . . , 𝑛2𝐿

𝑛
𝑛1, . . . , 𝑛2𝐿

(cid:19) 2𝐿
(cid:214)

𝑖=1
(cid:19) 2𝐿
(cid:214)

𝑖=1

∥𝑑𝑛𝑖

𝜏 𝑒 𝐴𝑖,ℓ ∥

(Λ𝑖ℓ/2𝑘)𝑛𝑖 (𝑛𝑖 + 2𝑥𝑖,ℓ,𝑛𝑖 )𝑛𝑖 .

(5.69)

(5.70)

(5.71)

Maximize Λ𝑖 over all 𝑖 = 1, . . . , 𝐿 and call it Λ. We can factor out the corresponding term, and

with some rewriting obtain

(Λℓ/2𝑘)𝑛 ∑︁

𝑁

(cid:18)

𝑛
𝑛1, . . . , 𝑛2𝐿

(cid:19) 2𝐿
(cid:214)

𝑖=1

(𝑛𝑖 + 2𝑥ℓ,𝑛𝑖 )𝑛𝑖

(5.72)

where we’ve also let 𝑥ℓ,𝑛𝑖 be 𝑥𝑖,ℓ,𝑛𝑖 with the subscript dropped on Λ𝑖. Focusing on the rightmost

product over 𝑖, one can show using a Lagrange multiplier that the maximum is given by 𝑛𝑖 = 𝑛/2𝐿

for all 𝑖 (we maximize over 𝑛𝑖 ∈ R+, which is an upper bound). This is intuitive from symmetry of

the product as well. Taking this as an upper bound, we have

+

2Λ𝜏
𝑘

+

𝑛
𝐿ℓ

(cid:19) 𝑛

(cid:18)

∑︁

𝑁

(cid:19)

𝑛
𝑛1, . . . , 𝑛2𝐿

∥𝑑𝑛

𝜏𝑈2(𝜏ℓ, 𝜏ℓ−1)∥ < (Λℓ/2𝑘)𝑛

= (Λℓ/2𝑘)𝑛

(cid:18) 𝑛
2𝐿

(cid:18)

𝑛 +

4Λ𝜏𝐿
𝑘

(cid:19) 𝑛

(cid:19) 𝑛

.

+

2𝑛
ℓ
2Λ𝜏𝐿ℓ
𝑘

(cid:18)

= (Λ/𝑘)𝑛

𝑛 + 𝑛ℓ/2 +

(5.73)

where in going to the second line we evaluated the multinomial sum as (2𝐿)𝑛 and simplified.

123

With this in hand, we return to (5.57) and bound it as

∥𝑑𝑛

𝜏𝑈 (𝑘)

2

(𝜏, 0)∥ ≤

(cid:18)

∑︁

𝑁

𝑛
𝑛1, . . . , 𝑛𝑘

(cid:19)

𝑘
(cid:214)

∥𝑑𝑛ℓ

𝜏 𝑈2(𝜏ℓ, 𝜏ℓ−1) ∥

𝑛
𝑛1, . . . , 𝑛𝑘
Using the upper bound ℓ ≤ 𝑘 and factoring out the (Λ/𝑘)𝑛ℓ using the sum rule,

𝑛ℓ + 𝑛ℓℓ/2 +

< ∑︁
𝑁

(Λ/𝑘)𝑛ℓ

2Λ𝜏𝐿ℓ
𝑘

ℓ=1

(cid:18)

(cid:18)

ℓ=1
𝑘
(cid:214)

(cid:19)

(5.74)

(cid:19) 𝑛ℓ

.

∥𝑑𝑛

𝑛
𝑛1, . . . , 𝑛𝑘
Similar to, we upper bound the product using 𝑛ℓ = 𝑛/𝑘 for all ℓ, which can be justified through a

(𝜏, 0)∥ < (Λ/𝑘)𝑛 ∑︁

(𝑛ℓ + 𝑛ℓ 𝑘/2 + 2Λ𝜏𝐿)𝑛ℓ .

𝜏𝑈 (𝑘)

(5.75)

ℓ=1

𝑁

2

(cid:18)

(cid:19)

𝑘
(cid:214)

maximization using Lagrange multipliers. The corresponding bound is

∥𝑑𝑛

𝜏𝑈 (𝑘)

2

(𝜏, 0)∥ < (Λ/𝑘)𝑛 (𝑛/𝑘 + 𝑛/2 + 2Λ𝜏𝐿)𝑛 ∑︁

𝑁

(cid:19)

(cid:18)

𝑛
𝑛1, . . . , 𝑛𝑘

= (𝑛Λ)𝑛

(cid:18) 1
𝑘

+

1
2

+

2Λ𝜏𝐿
𝑛

(cid:19) 𝑛

.

(5.76)

We are finally ready to return to equation (5.56) and bound 𝑅2𝑚. We recall that Λ has 𝜏

dependence, and let Λmax := max𝜏∈[0,𝑡] Λ(𝜏). We also upper bound any appearance of 𝜏 otherwise

by 𝑡 because these are always in the numerator. So far, using 𝑛 = 2𝑚 + 1, these reductions give

∥𝑅2𝑚 ∥ <

∥𝑎∥1
(2𝑚 + 1)!

((2𝑚 + 1)Λmax𝑡)2𝑚+1 max

𝑗

(cid:18) 1
𝑘 𝑗

+

1
2

+

2Λ𝑡 𝐿
2𝑚 + 1

(cid:19) 2𝑚+1

.

(5.77)

Employing a Stirling bound on the factorial, and factoring out an additional 𝐿 from the rightmost

term,

∥𝑅2𝑚 ∥ <

∥𝑎∥1
√︁2𝜋(2𝑚 + 1)

(𝑒𝐿Λmax𝑡)2𝑚+1 max

𝑗

(cid:18) 1
𝐿𝑘 𝑗

+

1
2𝐿

+

2Λ𝑡
2𝑚 + 1

(cid:19) 2𝑚+1

.

(5.78)

We now apply the assumption that 𝑒𝐿Λmax𝑡 < 1 to upper bound the max 𝑗 term, along with

𝑘 𝑗 , 𝐿 ≥ 1.

Thus,

(cid:18) 1
𝐿𝑘 𝑗

+

1
2𝐿

+

2Λ𝑡
2𝑚 + 1

max
𝑗

(cid:19) 2𝑚+1

(cid:19) 2𝑚+1

<

(cid:18) 3
2

+

2
3𝑒

∥𝑅2𝑚 ∥ <

<

∥𝑎∥1
√
𝜋𝑚
2
∥𝑎∥1
√
𝜋𝑚
2

(cid:18)(cid:18) 3𝑒
2

+

(cid:19)

2
3

𝐿 max
𝜏∈[0,𝑡]

Λ2𝑚+1(𝜏)𝑡

(cid:18)

5𝐿 max
𝜏∈[0,𝑡]

Λ2𝑚+1(𝜏)𝑡

(cid:19) 2𝑚+1

.

(cid:19) 2𝑚+1

(5.79)

(5.80)

In these last lines, we remind ourselves that Λ has the subscript 2𝑚 + 1 as per Definition 3.

□

124

5.5 Time Step Analysis

The next ingredient we need for a complexity analysis is asymptotic bounds on the number of

subintervals 𝑟 needed in the time mesh. This will be the concern of this section. Unfortunately, in

pursuing best-case bounds on 𝑟, we eschew a practical procedure for generating the time points 𝑡𝑖.

Section 5.9 provides a concrete procedure which is based on the analysis of this section.

For time dependent Hamiltonians, because the cost per unit time can vary with 𝑡 in general,

one should adaptively choose the step size depending on the cost. For our purposes, this means

choosing a step size inversely proportional to the energy measure Λ2𝑚+1(𝑡). We will explore this

adaptive time stepping and show 𝐿1-norm scaling with Λ2𝑚+1(𝑡) here.

To derive bounds on 𝑟, we will need to assume something about size of the derivative (cid:164)Λ2𝑚+1

compared to Λ2𝑚+1 itself. Given a Λ𝑛-bound, a differentiable (smooth, even) Λ𝑛-bound exists. From
now on, we consider Λ𝑛-bounds for which there exists a 𝐾 ∈ R+ be such that | (cid:164)Λ𝑛 (𝑡)| ≤ 𝐾Λ𝑛 (𝑡)2 for

all 𝑡 ∈ [0, 𝑇]. Given 𝐻 that is Λ𝑛 boundable, there is always, in fact, a Λ𝑛 bound such that 𝐾 exists

and is arbitrarily close to zero. For example, we may take a constant bound Λ′

𝑛 := max𝑡 Λ𝑛 (𝑡),

noting that Λ𝑛 is continuous on a compact interval. Of course, Λ′ does not capture the changing

behavior of 𝐻 (𝑡), and is therefore suboptimal. Nevertheless, we’ve demonstrated that our additional

assumptions are not much more restrictive than those we’ve already made. Note that (in natural

units) 𝐾 is dimensionless.

With these preliminaries in place, the following result provides an upper bound on the number

of time steps needed for our MPF algorithm.

Lemma 5.5.1. Let 𝐻 satisfy the assumptions of Theorem 5.4.1, and let Λ2𝑚+1 be a Λ2𝑚+1-bound for
𝐻 such that, for some 𝐾 ∈ R+, | (cid:164)Λ2𝑚+1(𝑡)| ≤ 𝐾Λ2𝑚+1(𝑡)2 for all 𝑡 ∈ [0, 𝑇]. For every 𝜖 > 0, there

exists a list (𝑡0, 𝑡1, . . . , 𝑡𝑟) of monotonically increasing times 𝑡 𝑗 ∈ [0, 𝑇], with 𝑡0 = 0 and 𝑡𝑟 = 𝑇,

such that

∥𝑈 (𝑇, 0) −

𝑟
(cid:214)

𝑖=1

𝑈2,𝑚 (𝑡𝑖, 𝑡𝑖−1) ∥ ≤ 𝜖

125

with the number of time steps 𝑟 bounded above as

(cid:36)(cid:18)

(cid:18)

5

1 +

𝑟 ≤

(cid:19)

𝐾

3
2

𝐿∥Λ∥1

(cid:19) 1+ 1

2𝑚 (cid:18) ∥𝑎∥1
√
𝜋𝑚
𝜖

2𝑚 (cid:37)
(cid:19) 1

.

Here, ∥Λ2𝑚+1∥1 is the 𝐿1 norm.

∫ 𝑇

∥Λ2𝑚+1∥1 :=

Λ2𝑚+1(𝑡)𝑑𝑡

0
Proof. As discussed in Section 5.4 in order to satisfy the 𝜖-error constraint of Lemma 5.5.1, it

suffices that the error on each subinterval is less than 𝜖/𝑟. Using Theorem 5.4.1, the sum is bounded

as

𝑟
∑︁

𝑖=1

∥𝑈 (𝑡𝑖, 𝑡𝑖−1) − 𝑈2,𝑚 (𝑡𝑖, 𝑡𝑖−1)∥ ≤

∥𝑎∥1√
𝜋𝑚

𝑟
∑︁

(cid:18)

𝑖=1

5𝐿 max

𝜏∈[𝑡𝑖−1,𝑡𝑖]

Λ2𝑚+1(𝜏) (𝑡𝑖 − 𝑡𝑖−1)

(cid:19) 2𝑚+1

.

(5.81)

To ensure an overall error 𝜖, it therefore suffices to produce a mesh such that for each 𝑖,

(cid:18)

∥𝑎∥1√
𝜋𝑚

5𝐿 max

𝜏∈[𝑡𝑖−1,𝑡𝑖]

Λ2𝑚+1(𝜏) (𝑡𝑖 − 𝑡𝑖−1)

(cid:19) 2𝑚+1

≤ 𝜖/𝑟.

Rearranging, this corresponds to choosing 𝑡𝑖, given all other parameters, that satisfy

𝐿 max

𝜏∈[𝑡𝑖−1,𝑡𝑖]

Λ2𝑚+1(𝜏) (𝑡𝑖 − 𝑡𝑖−1) ≤

√

(cid:18) 𝜖
𝜋𝑚
∥𝑎∥1𝑟

1
5

(cid:19) 1/(2𝑚+1)

.

(5.82)

(5.83)

We now digress in order to relate max𝜏 Λ2𝑚+1(𝜏) and its average. Here is where we will make
use of the 𝐾-bounds on the derivative (cid:164)Λ, we closely follow arguments found in [137]. From the

inequality in the lemma statement, we have

(cid:164)Λ2𝑚+1(𝑡)
Λ2𝑚+1(𝑡)2
1
Λ2𝑚+1(𝑡)
Suppose the time 𝑡𝑖−1 has been chosen by the previous iteration (if 𝑖 = 1, 𝑡0 = 0). Let 𝑡 > 𝑡𝑖−1 and

(5.84)

≤ 𝐾.

≤ 𝐾

(cid:12)
(cid:12)
(cid:12)
(cid:12)
𝑑
𝑑𝑡

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

integrate the above inequality from 𝑡𝑖−1 to 𝑡.

∫ 𝑡

𝑡𝑖−1
∫ 𝑡

(cid:12)
(cid:12)
(cid:12)
(cid:12)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑
𝑑𝜏
𝑑
𝑑𝜏

(cid:12)
(cid:12)
(cid:12)
(cid:12)

1
Λ2𝑚+1(𝜏)
1
Λ2𝑚+1(𝜏)

𝑑𝜏

−

1
Λ2𝑚+1(𝑡𝑖−1)

126

𝑡𝑖−1
1
Λ2𝑚+1(𝑡)

(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝜏 ≤ 𝐾 (𝑡 − 𝑡𝑖−1)

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤ 𝐾 (𝑡 − 𝑡𝑖−1)

≤ 𝐾 (𝑡 − 𝑡𝑖−1)

(5.85)

Let us rearrange this in terms of Λ2𝑚+1(𝑡) alone.

−𝐾 (𝑡 − 𝑡𝑖−1) ≤

1
Λ2𝑚+1(𝑡𝑖−1)

− 𝐾 (𝑡 − 𝑡𝑖−1) ≤

Λ2𝑚+1(𝑡𝑖−1)
1 + 𝐾 (𝑡 − 𝑡𝑖−1)Λ2𝑚+1(𝑡𝑖−1)

−

1
Λ2𝑚+1(𝑡)
1
Λ2𝑚+1(𝑡)
≤ Λ2𝑚+1(𝑡) ≤

≤

1
Λ2𝑚+1(𝑡𝑖−1)
1
Λ2𝑚+1(𝑡𝑖−1)

Λ2𝑚+1(𝑡𝑖−1)
1 − 𝐾 (𝑡 − 𝑡𝑖−1)Λ2𝑚+1(𝑡𝑖−1)

≤ 𝐾 (𝑡 − 𝑡𝑖−1)

+ 𝐾 (𝑡 − 𝑡𝑖−1)

(5.86)

The lowerbound inequality holds for all 𝑡 > 𝑡𝑖−1, while the upper bound only holds when

(𝑡 − 𝑡𝑖−1)Λ2𝑚+1(𝑡𝑖−1)𝐾 < 1.

(5.87)

We restrict our attention to 𝑡 for which both bounds hold. Consider, for the moment, only the

leftmost inequality. The lower bound on the left is monotonically decreasing with 𝑡. This means

that it is also a uniform lower bound on Λ2𝑚+1(𝑡′) for any 𝑡′ ∈ [𝑡𝑖−1, 𝑡]. Therefore, it is a lower

bound for the average ¯Λ2𝑚+1(𝑡) on the interval [𝑡𝑖−1, 𝑡].

That is,

¯Λ2𝑚+1(𝑡, 𝑡𝑖−1) :=

1
𝑡 − 𝑡𝑖−1

∫ 𝑡

𝑡𝑖−1

Λ2𝑚+1(𝜏)𝑑𝜏

Λ2𝑚+1(𝑡𝑖−1)
1 + 𝐾 (𝑡 − 𝑡𝑖−1)Λ2𝑚+1(𝑡𝑖−1)

≤ ¯Λ2𝑚+1(𝑡, 𝑡𝑖−1),

or, after isolating for Λ2𝑚+1(𝑡𝑖−1)

Λ2𝑚+1(𝑡𝑖−1) ≤

¯Λ2𝑚+1(𝑡, 𝑡𝑖−1)
1 − 𝐾 (𝑡 − 𝑡𝑖−1) ¯Λ2𝑚+1(𝑡, 𝑡𝑖−1)

.

(5.88)

(5.89)

(5.90)

At this point, let’s now consider the upper bound in equation (5.86). This bound is monotonically

increasing in 𝑡, and therefore also upper bounds Λ2𝑚+1(𝜏) for any 𝜏 in [𝑡𝑖−1, 𝑡]. Therefore, it is also

a bound for the maximum.

max
𝜏∈[𝑡𝑖−1,𝑡]

Λ2𝑚+1(𝜏) ≤

Λ2𝑚+1(𝑡𝑖−1)
1 − 𝐾 (𝑡 − 𝑡𝑖−1)Λ2𝑚+1(𝑡𝑖−1)

(5.91)

Substituting bounds for Λ2𝑚+1(𝑡𝑖−1) from equation (5.90) gives us a bound on the maximum value

in terms of the average.

max
𝜏∈[𝑡𝑖−1,𝑡]

Λ2𝑚+1(𝜏) ≤

¯Λ2𝑚+1(𝑡, 𝑡𝑖−1)

1 − 3
2

𝐾 ¯Λ2𝑚+1(𝑡, 𝑡𝑖−1) (𝑡 − 𝑡𝑖−1)

(5.92)

127

Solving for the average value of Λ2𝑚+1, and multiplying by 𝑡 − 𝑡𝑖−1 on both sides,

(𝑡 − 𝑡𝑖−1) ¯Λ2𝑚+1(𝑡, 𝑡𝑖−1) ≥

(𝑡 − 𝑡𝑖−1) max𝜏∈[𝑡𝑖−1,𝑡] Λ2𝑚+1(𝜏)
𝐾 (𝑡 − 𝑡𝑖−1) max𝜏∈[𝑡𝑖−1,𝑡] Λ2𝑚+1(𝜏)

1 + 3
2

.

(5.93)

Let us finally choose a 𝑡 = 𝑡𝑖 which will serve as the next time step in the adaptive scheme. We

would like come as close as possible to saturating equation (5.83) while staying within the constraint

imposed by the maximum bound of equation (5.86). Thus, we choose 𝑡𝑖 such that

max
𝜏∈[𝑡𝑖−1,𝑡𝑖]

Λ2𝑚+1(𝜏)(𝑡𝑖 − 𝑡𝑖−1) = min

(cid:40)

1
𝐾

, 1
5𝐿

√

(cid:18) 𝜖
𝜋𝑚
∥𝑎∥1𝑟

(cid:19) 1/(2𝑚+1)(cid:41)

.

(5.94)

Since 𝐾 is a constant, for asymptotic purposes we will assume sufficiently small 𝜖 such that the

right term is smaller. Plugging in to (5.93) yields

¯Λ2𝑚+1(𝑡𝑖, 𝑡𝑖−1)(𝑡𝑖 − 𝑡𝑖−1) ≥

1
5𝐿

(cid:17) 1/(2𝑚+1)

(cid:16) 𝜖

√

𝜋𝑚
∥𝑎∥1𝑟
√
(cid:16) 𝜖

𝜋𝑚
∥𝑎∥1𝑟

(cid:17) 1/(2𝑚+1)

.

𝐾

(5.95)

< 1 and by summing over 𝑖 = 1, . . . , 𝑟

1 + 3
2

1
5𝐿

(cid:17) 1/(2𝑚+1)

(cid:16) 𝜖

√

𝜋𝑚
∥𝑎∥1𝑟

We then find, by using the fact that 1
5𝐿

in (5.95), that

∥Λ∥1 ≥ 𝑟 2𝑚

2𝑚+1

1
5𝐿

(cid:18) 𝜖

√

𝜋𝑚
∥𝑎∥1

(cid:19) 1/(2𝑚+1) (cid:18)

1 +

(cid:19) −1

.

𝐾

3
2

(5.96)

Finally, rearranging the above, this implies that the number of steps required for the MPF algorithm

is upper bounded as

(cid:18)

(cid:18)

5

1 +

𝑟 ≤

(cid:19)

𝐾

3
2

𝐿∥Λ∥1

(cid:19) 1+ 1

2𝑚 (cid:18) ∥𝑎∥1
√
𝜋𝑚
𝜖

(cid:19) 1
2𝑚

.

The result then directly follows from the requirement that 𝑟 is an integer.

(5.97)

□

To summarize, we’ve provided an upper bound on the number of steps 𝑟 needed given as-

sumptions on the derivative of Λ2𝑚+1. What is perhaps objectionable is that, in determining our

subsequent time stepping, we seemed to need information about the total number of steps 𝑟 that we

would end up with. While this does not detract from the correctness of our result, it does indicate

possible difficulty in constructing a suitable set of 𝑡 𝑗 for which the Lemma holds. One approach is

to guess the final number 𝑟try of steps needed, construct the mesh according to the proof, then see

if 𝑟try can be made correct. This approach is considered in Section 5.9.

128

5.6 Query Complexity

With the results of the previous two sections, we proceed to bound the query complexity needed

to perform a time dependent MPF simulation. First, we define a set of oracles that are appropriate

for this simulation problem. We reemphasize that the most natural input model in our setting is the

linear combinations of Hamiltonians model

𝐻 =

𝐿
∑︁

𝑗=1

𝛼 𝑗 (𝑡)𝐻 𝑗 ,

(5.98)

where 𝛼 𝑗 : [0, 𝑇] → R has 2𝑚 + 1 continuous derivatives and 𝐻 𝑗 ∈ Herm(C2𝑛
We discretize [0, 𝑇] into 2𝑛𝑡 uniform grid points 𝑡𝑘 = 𝑘𝑇/2𝑛𝑡 for 𝑘 ∈ [0, 2𝑛𝑡 ) ∩ Z, and define

) satisfies ∥𝐻 𝑗 ∥ ≤ 1.

𝛼 𝑗 𝑘 := 𝛼 𝑗 (𝑡𝑘 ). Let 𝛿𝑡 := 𝑇/2𝑛𝑡 . Let 𝑈𝛼 and 𝑈𝐻 be unitary oracles which provide the input

Hamiltonian as follows.

𝑈𝛼 | 𝑗⟩ |𝑘⟩ |𝜏⟩ |0⟩ := | 𝑗⟩ |𝑘⟩ |𝜏⟩ (cid:12)
(cid:12)𝛼 𝑗 𝑘 𝜏(cid:11) |𝜓⟩ := | 𝑗⟩ (cid:12)

𝑈𝐻 | 𝑗⟩ (cid:12)

(cid:12)𝛼 𝑗 𝑘 𝜏(cid:11)

(cid:12)𝛼 𝑗 𝑘 𝜏(cid:11) exp{−𝑖𝐻 𝑗 𝛼 𝑗 𝑘 𝜏} |𝜓⟩

(5.99)

The oracle 𝑈𝛼 encodes a reversible classical computation and may be taken as self-inverse. Here

|𝜏⟩ encodes a step of size 𝜏 ∈ R in binary using 𝑛𝑐 qubits. Such step sizes are always nonnegative

for the low-order formulas we consider, and therefore we take 𝜏 ∈ [0, 𝑇]. Hence, 𝛿𝑡 = 𝑇/2𝑛𝑐 is the

rounding error for the step sizes. We neglect rounding effects due to the values 𝛼 𝑗 𝑘 𝜏.

Our first result concerns the approximate implementation of 𝑈2 using the two oracles.

Lemma 5.6.1. Let 𝑈2(𝜏 + 𝑡, 𝑡) be the 2nd-order Suzuki-Trotter formula for the midpoint formula of

equation (5.8). Then an approximation 𝑊2 can be constructed using at most 6𝐿 − 3 queries to 𝑈𝐻

and 𝑈𝛼, such that

∥𝑈2(𝑡 + 𝜏, 𝑡) − 𝑊2(𝑡 + 𝜏, 𝑡) ∥ ≤ 𝐿 max
𝑗,𝑡∈[0,𝑇]

| (cid:164)𝛼 𝑗 (𝑡 + 𝜏/2)|

𝑇 2
2𝑛𝑐

.

Proof. Define 𝑊2 as 𝑈2 but with each 𝛼 𝑗 evaluated at the nearest discrete times in {𝑡𝑘 }. Using the

techniques of [137], two queries to 𝑈𝛼 and one query to 𝑈𝐻 are needed to exactly simulate each of

the 2𝐿 − 1 exponentials present in 𝑊2. Thus 3 × (2𝐿 − 1) queries are needed total. To evaluate the

129

discretization error, by Box 4.1 of [101] we have that

∥𝑊2 − 𝑈2∥ ≤ 2

𝐿
∑︁

𝑗=1

∥𝑒−𝑖𝐻 𝑗 𝛼 𝑗 (rnd[𝑡+𝜏/2])𝜏/2 − 𝑒−𝑖𝐻 𝑗 𝛼 𝑗 (𝑡+𝜏/2)𝜏/2∥

(5.100)

which in turn is upper bounded, through an application of the fundamental theorem of calculus, by

2

𝐿
∑︁

𝑗=1

(cid:13)
(cid:13)𝐻 𝑗 𝛼 𝑗 (rnd (cid:2)𝑡 +

(cid:3))

𝜏

2

𝜏

2

− 𝐻 𝑗 𝛼 𝑗 (𝑡 +

𝜏

2

)

𝜏

2

(cid:13)
(cid:13)

(5.101)

where rnd rounds to the nearest 𝑛𝑐-bit value. Since ∥𝐻 𝑗 ∥ ≤ 1 this is merely upper bounded as

𝜏

𝐿
∑︁

𝑗=1

(cid:12)
(cid:12)𝛼 𝑗 (rnd (cid:2)𝑡 +

𝜏

2

(cid:3)) − 𝛼 𝑗 (𝑡 +

𝜏

2

)(cid:12)
(cid:12).

(5.102)

By the fundamental theorem of calculus, with an integral upper bound, each term is upper bounded

as 𝛿𝑡 max𝛿𝑡∈𝑡±𝛿𝑡 |𝜕𝑡𝛼 𝑗 (𝑡 + 𝜏/2)|. Maximizing over [0, 𝑇] instead, and making other simplifying

choices,we get a crude upper bound

∥𝑊2 − 𝑈2∥ ≤ 𝜏𝐿𝛿𝑡 max
𝑗,[0,𝑇]

| (cid:164)𝛼 𝑗 (𝑡)|

≤ 𝐿

𝑇 2
2𝑛𝑐

| (cid:164)𝛼 𝑗 (𝑡)|.

max
𝑗,[0,𝑇]

Rearranging this gives the inequality of the lemma statement.

(5.103)

□

Having supplied an approximate base formula 𝑊2 with our queries, we next need to implement

an approximate MPF 𝑊2,𝑚 over a subinterval [𝑡0, 𝑡1]. This is conventionally done through the use

of "select" SEL and "prepare" PREP circuits.

√︄

𝑚
∑︁

PREP |0⟩ :=

|𝑎 𝑗 |
∥𝑎∥1

| 𝑗⟩

𝑗=1
SEL | 𝑗⟩ |𝜓⟩ := sgn(𝑎 𝑗 ) | 𝑗⟩ 𝑊 (𝑘 𝑗 )

(𝑡1, 𝑡0) |𝜓⟩

(5.104)

2

The circuit PREP can be implemented without any queries to 𝑈𝛼 or 𝑈𝐻 whereas SEL requires

𝑂 (𝐿∥ (cid:174)𝑘 ∥∞) queries. Following the well-conditioned MPF scheme of [90] we have that 𝑘 𝑗 ≤ 3𝑚2.

This implies that a query to SEL requires 𝑂 (𝐿𝑚2) queries to 𝑈𝐻 and 𝑈𝛼.

We can use the SEL and PREP for a standard LCU block encoding in order to construct a time

dependent MPF with base formula 𝑊2.

130

Lemma 5.6.2. Under the assumptions of Theorem 5.4.1 and the query model above, for any

[𝑡0, 𝑡1] ⊆ [0, 𝑇] the time dependent MPF 𝑊2,𝑚 with base formula 𝑊2 satisfies

∥𝑊2,𝑚 (𝑡1, 𝑡0) − 𝑈 (𝑡1, 𝑡0) ∥ ∈ 𝑂

∥𝑎∥1

(cid:32)

(cid:18)

max
𝑡∈[𝑡0,𝑡1]

Λ2𝑚+1(𝑡)𝑇

(cid:19) 2𝑚+1(cid:33)

,

provided that

(cid:32)

𝑛𝑐 ≥ log

√

𝜋𝑚5/2𝐿 max𝑡, 𝑗 |𝜕𝑡𝛼 𝑗 (𝑡)|(𝑡1 − 𝑡0)2
3
(cid:0)5𝐿 max𝑡∈[𝑡0,𝑡1] Λ2𝑚+1(𝑡) (𝑡1 − 𝑡0)(cid:1) 2𝑚+1

(cid:33)

,

and can be constructed with a number of queries to 𝑈𝐻 and 𝑈𝛼 scaling as 𝑂 (𝑚2𝐿).

Proof. From Lemma 4 of [15], we have

(⟨0| ⊗ 𝐼)(PREP†)SEL(PREP) (|0⟩ ⊗ 𝐼) =

1
∥𝑎∥1

𝑚
∑︁

𝑗=1

𝑎 𝑗𝑊 (𝑘 𝑗 )
2

(𝑡1, 𝑡0)

Let 𝛿′ > 0 be such that, for all 𝑗 and ℓ ∈ {1, . . . , 𝑘 𝑗 },
(cid:18)

(cid:19)

(cid:18)

(cid:13)
(cid:13)
𝑊2
(cid:13)
(cid:13)

Δ𝑡

ℓ
𝑘 𝑗

+ 𝑡0, Δ𝑡

+ 𝑡0

− 𝑈2

Δ𝑡

ℓ − 1
𝑘 𝑗

= 𝑊2,𝑚 (𝑡1, 𝑡0)/∥𝑎∥1.

ℓ
𝑘 𝑗

+ 𝑡0, Δ𝑡

ℓ − 1
𝑘 𝑗

+ 𝑡0

(cid:19)(cid:13)
(cid:13)
(cid:13)
(cid:13)

≤ 𝛿′

where Δ𝑡 = 𝑡1 − 𝑡0. Then, by invoking Box 4.1 from [101],

(cid:13)
𝑈 (𝑘 𝑗 )
(cid:13)
(cid:13)
2

(𝑡1, 𝑡0) − 𝑊 (𝑘 𝑗 )

2

(𝑡1, 𝑡0)

(cid:13)
(cid:13)
(cid:13)

≤ 𝑘 𝑗 𝛿′

which, since 𝑘 𝑗 ≤ 3𝑚2, implies that

(5.105)

(5.106)

(5.107)

∥𝑉2,𝑚 (𝑡1, 𝑡0) − 𝑊2,𝑚 (𝑡1, 𝑡0) ∥ ≤ 3𝑚2𝛿′∥𝑎∥1.

(5.108)

We supply 𝛿′ using Lemma 5.6.1, obtaining

3𝑚2𝛿′∥𝑎∥1 ≤ 3𝑚2∥𝑎∥1𝐿 max
𝑗,𝑡∈[0,𝑇]

| (cid:164)𝛼 𝑗 (𝑡 + 𝜏/2)|

𝑇 2
2𝑛𝑐

,

(5.109)

giving us a bound on the discretized MPF 𝑊2,𝑚 relative to the undiscretized 𝑉2,𝑚.

It then follows from the triangle inequality and Theorem 5.4.1 that

(cid:13)𝑊2,𝑚 (𝑡1, 𝑡0) − 𝑈 (𝑡1, 𝑡0)(cid:13)
(cid:13)

(cid:13) ≤ (cid:13)

(cid:13) + ∥𝑊2,𝑚 (𝑡1, 𝑡0) − 𝑈2,𝑚 (𝑡1, 𝑡0) ∥

(cid:13)𝑈2,𝑚 (𝑡1, 𝑡0) − 𝑈 (𝑡1, 𝑡0)(cid:13)
∥𝑎∥1√
𝜋𝑚

5𝐿 max
𝑡∈[0,𝑇]

Λ2𝑚+1(𝑡)𝑇

(cid:18)

≤

(cid:19) 2𝑚+1

+ 3𝑚2𝐿∥𝑎∥1 max
𝑗,𝑡

| (cid:164)𝛼 𝑗 (𝑡)|

𝑇 2
2𝑛𝑐

.

(5.110)

131

Under the assumption that

𝑛𝑐 ≥ log

(cid:32)

√
𝜋𝑚5/2𝐿 max 𝑗,𝑡 | (cid:164)𝛼 𝑗 (𝑡)|𝑇 2
3
(cid:0)5𝐿 max𝑡∈[0,𝑇] Λ2𝑚+1(𝜏)𝑇 (cid:1) 2𝑚+1

(cid:33)

(5.111)

the second term is bounded by the first (5.110), so we have an upper bound

𝑎 𝑗

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

𝑀
∑︁

𝑘 𝑗
(cid:214)

(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)
Since 𝑈 (𝑇, 0) is unitary, we know that the MPF implemented by our algorithm is close to a unitary.

(cid:0)𝑇 𝑞/𝑘 𝑗 , 𝑇 (𝑞 − 1)/𝑘 𝑗 (cid:1) − 𝑈 (𝑇, 0)

5𝐿 max
𝑡∈[0,𝑇]

2∥𝑎∥1√
𝜋𝑚

Λ2𝑚+1(𝑡)𝑇

(5.112)

(cid:19) 2𝑚+1

𝑊2

𝑞=1

𝑗=1

≤

(cid:18)

.

This means that we satisfy the preconditions of robust oblivious amplitude amplification given

by Lemma 5 of [15]. This result implies that using 𝑂 (∥𝑎∥1) applications of the unitary given
by (5.105), we can implement an operator (cid:101)𝑊 (𝑇, 0) such that (for constant 𝑚)

∥𝑊2,𝑚 (𝑡1, 𝑡0) − 𝑈 (𝑡1, 𝑡0)∥ ∈ 𝑂

∥𝑎∥1

(cid:32)

(cid:18)

max
𝑡∈[0,𝑇]

Λ2𝑚+1(𝑡) (𝑡1 − 𝑡0)

(cid:19) 2𝑚+1(cid:33)

.

The number of queries scales as

𝑄step ∈ 𝑂 (∥𝑎∥1𝑚2𝐿) ⊆ (cid:101)𝑂 (𝑚2𝐿).

(5.113)

(5.114)

□

With the short-time simulation costs in place we are now ready to state our main theorem,

which bounds the number of queries needed to perform the full multiproduct simulation of a time

dependent Hamiltonian.

Theorem 5.6.3. In the query setting above, and under the assumptions of Theorem 5.4.1, and

Lemma 5.5.1 (Λ2𝑚+1-bounded 𝐻 with 𝐾 bound on (cid:164)Λ2𝑚+1), we have that the number of queries 𝑄tot

needed to 𝑈𝛼 and 𝑈𝐻 to construct an operator 𝑊tot(𝑇, 0) simulate a time dependent Hamiltonian
of the form (cid:205)𝐿

𝛼 𝑗 (𝑡)𝐻 𝑗 such that ∥(⟨0| ⊗ 𝐼)𝑊tot(𝑇, 0) (|0⟩ ⊗ 𝐼) − 𝑈 (𝑇, 0) ∥ ≤ 𝜖 satisfies

𝑗=1

(cid:16)

𝑄tot ∈ (cid:101)𝑂

𝐿(1 + 𝐾) ∥Λ2𝑚+1∥1 log2(1/𝜖)

(cid:17)

,

and the total number of auxiliary qubits is in

(cid:32)

(cid:101)𝑂

log

(cid:32) 𝐿 (1 + 𝐾) ∥Λ2𝑚+1∥1 max 𝑗,𝑡 | (cid:164)𝛼 𝑗 (𝑡)|𝑇 2
𝜖

(cid:33)(cid:33)

.

132

Proof. From Lemma 5.5.1 we have that the number of segments needed to perform the simulation

within error 𝜖 obeys

𝑟 ∈ (cid:101)𝑂

(cid:18) ((1 + 𝐾) ∥Λ2𝑚+1∥1)1+1/(2𝑚)
𝜖 1/(2𝑚)

(cid:19)

.

Therefore, using Lemma 5.6.2,

𝑄tot ∈ (cid:101)𝑂 (𝑚2𝐿𝑟) ⊆ (cid:101)𝑂

(cid:18) 𝑚2𝐿 ((1 + 𝐾) ∥Λ2𝑚+1∥1)1+1/(2𝑚)
𝜖 1/(2𝑚)

(cid:19)

.

(5.115)

(5.116)

An approximation to the optimal value of 𝑚 ∈ Z+ can be obtained by equating the exponentially

shrinking component of the cost to the polynomially increasing value of 𝑚. We choose 𝑚 to satisfy
(cid:18) (1 + 𝐾) ∥Λ2𝑚+1∥1
𝜖

(5.117)

𝑚2 =

(cid:19) 1/2𝑚

.

Solving for 𝑚 yields

𝑚 =

log (cid:0)(1 + 𝐾)∥Λ2𝑚+1∥1/𝜖 (cid:1)

4 LambertW

(cid:16)

log (cid:0)(1 + 𝐾)∥Λ2𝑚+1∥1/𝜖 (cid:1)/4

(cid:17) ∈ (cid:101)𝑂

(cid:18)

log

(cid:18) (1 + 𝐾) ∥Λ2𝑚+1∥1
𝜖

(cid:19)(cid:19)

.

(5.118)

This implies that the query complexity 𝑄tot is in

(cid:16)

(cid:101)𝑂

𝐿(1 + 𝐾) ∥Λ2𝑚+1∥1 log2(1/𝜖)

(cid:17)

.

(5.119)

The number of auxiliary qubits needed in the construction is in 𝑂 (log(𝑚)) to implement the MPF

and (⌈log 𝐿⌉ + 𝑛𝑐) to implement the 𝑈𝛼 oracle. From the result of Lemma 5.6.2 we see that 𝑛𝑐

dominates this cost. We thus have a number of auxiliary qubits scaling as

(cid:32) 𝑚2𝐿 max |𝜕𝑡𝛼 𝑗 (𝑡)|𝑇 2
(cid:0)max𝑡∈[0,𝑇] Λ2𝑚+1(𝑡)𝑇 (cid:1) 2𝑚+1
(cid:32) 𝑚2𝐿𝑟 ∥𝑎∥1 max |𝜕𝑡𝛼 𝑗 (𝑡)|𝑇 2
𝜖

(cid:33)(cid:33)

(cid:33)(cid:33)

log

log

𝑛aux ∈ 𝑂

∈ 𝑂

(cid:32)

(cid:32)

(cid:32)

∈ (cid:101)𝑂

log

(cid:32) 𝐿 (1 + 𝐾) ∥Λ2𝑚+1∥1 max |𝜕𝑡𝛼 𝑗 (𝑡)|𝑇 2
𝜖

(cid:33)(cid:33)

where used Eq. (5.116) and Eq. (5.118) above.

(5.120)

□

This shows that the cost of quantum simulation using MPFs broadly conforms to the cost scalings

that one would expect of previous methods. In particular, similar to the truncated Dyson series

simulation method [89, 75] we obtain that the cost of simulating a time dependent Hamiltonian

scales near-linearly with time 𝑇 and poly-logarithmically with 1/𝜖.

133

5.7 Numerical Demonstrations

In the above sections, we developed and characterized MPFs for time dependent simulations by

showing their existence and proving error bounds. However, these bounds are unlikely to be the

final word on the performance of the algorithm. For example, we already mentioned that, for time

independent 𝐻, the MPF of Definition 2 is exact in cases where the Hamiltonian consists of only

commuting terms. Yet this behavior is not captured in the bound of Theorem 5.4.1 because Λ2𝑚

is at least as large as ∥𝐻 ∥. This discrepancy is unrelated to the fact that, in practice, the 2nd-order

formula 𝑈2 can only be computed approximately.

To begin bridging the gap between algorithm’s actual performance and our bounds, we inves-

tigate time dependent MPFs empirically through two numerical examples. We compute 𝑈2,𝑚 for

these systems on a classical computer (using matrix computations) and compare the result with the

exact propagator (computed within machine 𝜖). The vector (cid:174)𝑘 ∈ Z𝑚

+ we will use comes from the

bottom half of Table I from [90], which minimizes ∥ (cid:174)𝑘 ∥ for ∥𝑎∥1 ≤ 2.

In general, deriving an analytical solution for the propagator given a time dependent Hamilto-

nian is challenging or impossible. To bypass this problem, we will consider a time independent

Hamiltonian which is viewed from a "non-inertial" frame, thereby rendering the dynamics time

dependent in the new frame. More specifically, suppose 𝐻 is a time independent Hamiltonian with

propagator 𝑈 (𝑡) = 𝑒−𝑖𝐻𝑡 (henceforth the initial time is set to zero). Let |𝜓𝑡⟩ be the solution to

the Schrödinger equation 𝑖𝜕𝑡 |𝜓𝑡⟩ = 𝐻 |𝜓𝑡⟩. Under a frame transformation 𝑇 (𝑡), which transforms

vectors as | ˜𝜓𝑡⟩ = 𝑇 (𝑡) |𝜓𝑡⟩, the Hamiltonian and propagator transform as

˜𝑈 (𝑡) = 𝑇 (𝑡)𝑈 (𝑡)

˜𝐻 (𝑡) = 𝑖

𝜕𝑇 (𝑡)
𝜕𝑡

𝑇 (𝑡)† + 𝑇 (𝑡)𝐻 (𝑡)𝑇 (𝑡)†.

(5.121)

Thus, in order to benchmark the error of the MPF, we compute ˜𝑈𝑘 for Hamiltonian ˜𝐻, then compare

with the exact propagator (accurate to machine precision).

𝜖𝑐 = ∥ ˜𝑈2,𝑚 (𝑡) − 𝑇 (𝑡)𝑈 (𝑡) ∥

(5.122)

134

5.7.1 Example 1: Electron in Magnetic field, Rotating Frame

As a very simple first demonstration, consider a spin-1/2 particle (say, electron) in a homoge-

neous external magnetic field 𝐵. Choose a coordinate system such that 𝐵 makes an angle 𝜃 with

respect to the 𝑧-axis, and lies within the 𝑥𝑧 plane. This system can be described by the Hamiltonian

𝐻 = 𝜇𝐵(cos 𝜃𝑍/2 + sin 𝜃 𝑋/2)

(5.123)

where 𝑍 and 𝑋 (and later 𝑌 ) are Pauli operators, and 𝜇 is a coupling parameter that will henceforth

be set to one. The propagator 𝑈 (𝑡) = 𝑒−𝑖𝐻𝑡 is easy to compute, and corresponds to precession about

the magnetic field axis with frequency 𝐵.

To obtain a time dependent problem, let’s shift to a reference frame that rotates with angular

frequency 𝜔 about the 𝑧-axis. The transformation is given by 𝑅𝑧 (𝜔𝑡), where 𝑅𝑎 is the usual 𝑆𝑈 (2)

rotation operator about axis 𝑎. The Hamiltonian in the rotating frame is

˜𝐻 (𝑡) = (𝜔 + 𝐵 cos 𝜃)𝑍/2 + 𝐵 sin 𝜃 (cos 𝜔𝑡 𝑋/2 + sin 𝜔𝑡𝑌 /2).

(5.124)

Because we know that this Hamiltonian is just a transformed time independent system, it is easy to

compute the exact propagator ˜𝑈 (𝑡).

˜𝑈 (𝑡) = 𝑅𝑧 (𝜔𝑡)𝑈 (𝑡)

(5.125)

Though it is not strictly necessary to run the algorithm, let’s compute an appropriate Λ(𝑡) upper

bound. The spectral norm of ˜𝐻 may be upper bounded as

∥ ˜𝐻 ∥ ≤

|𝜔 + 𝐵 cos 𝜃|
2

+ |𝐵 sin 𝜃|

while the derivatives ˜𝐻 (𝑛) (𝑡) have the bound

∥ ˜𝐻 (𝑛) (𝑡) ∥ ≤ |𝐵 sin 𝜃𝜔𝑛|
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝐵 sin 𝜃
𝜔

(cid:12)
(cid:12)
(cid:12)
(cid:12)

∥𝑡𝑖𝑙𝑑𝑒𝐻 (𝑛) (𝑡) ∥ ≤ 𝜔

1/𝑛+1

𝑛+1√︃

(5.126)

(5.127)

.

For 𝜔 not too much larger than 𝐵, we see then that Λ(𝑡) = 𝜔 is an appropriate choice.

135

Figure 5.1 (left) Multiproduct errors plotted against simulation time, for several low-order MPFs,
on a log-log plot. Notice the power law scaling for small values of 𝑡. The parameters used here are
𝐵 = 1, 𝜔 = 4, 𝜃 = 𝜋/6. For larger 𝑚, one quickly runs into machine precision becoming the
dominant error source. (right) The running power 𝑝(𝑡, 𝑡′) defined in equation (5.128), with 𝑡′ = .3.
Note the plateau corresponds with the anticipated value of 2𝑚 + 1.

The first thing to check will be that the error has the appropriate power law scaling. Namely,

for 𝑀-term formulas, the error 𝜖𝑐 for small 𝑡 should scale as 𝑂 (𝑡2𝑚+1) or better. We can check this

by computing the "running power" 𝑝(𝑡, 𝑡′).

𝑝(𝑡, 𝑡′) :=

log 𝜖𝑡/𝜖𝑡′
log 𝑡/𝑡′

(5.128)

For different but small values of 𝑡, 𝑡′, the value of 𝑝 should approach the expected order of the

error: 2𝑚 + 1.

Indeed, this is precisely the behavior observed in Figure 5.1. For sufficiently

small simulation times, a power-law dependence on the simulation error is observed, and the

corresponding power is as anticipated. Additionally, we see that the error decreases by orders of

magnitude with each additional term once the power-law regime is reached. Choosing 𝑚 > 4 in

this example quickly leads to machine precision being the dominant error source.

Next, we vary the MPF order 𝑚 for fixed simulation time 𝑡. Since Λ = 𝜔, our bounds

predict an exponential decay in the error, but only provided 𝑡 < 1/𝜔. Otherwise, the bounds grow

exponentially and say nothing useful about performance. In Figure 5.7.1, we fix 𝑡 at several different

times and plot the error dependence on the multiproduct order 𝑚. Past a certain threshold value for

𝑚 (which increases with 𝑡) an exponential decay in error is observed, possibly superexponential.

It is promising that, even for 𝑡 = 10, the exponential decay is eventually achieved at 𝑚 ≳ 6.

136

Figure 5.2 Multiproduct error shows an (super)exponential decrease in error for sufficiently large
order 𝑚. The threshold for this regime is seen to increase as the simulation time increases. This
behavior surpasses the expectation of our proven bounds, since there are no guarantees if the time
step is too large. Note that, in practice, one should typically split a longer simulation time into
smaller steps. The plateau for 𝑡 = 1, 𝑚 > 8 is a result of machine precision limitations. Parameter
values: 𝐵 = 1, 𝜔 = 4, 𝜃 = 𝜋/6.

This suggests our error bounds may be too conservative, and in particular MPFs could absolutely

converge to 𝑈 as 𝑚 → ∞ in certain circumstances. This would be a notable improvement to

product formulas alone, which tend to lead to errors that diverge as 𝑚 → ∞ if the time step 𝑡

remains fixed [14, 137, 26]. In contrast, Theorem 5.4.1 shows that if the time step is sufficiently

small, then the MPF converges to the exact result. However, such convergence is not anticipated

from the bounds for a large value such as 𝑡 = 10.

Indeed, there are good reasons to believe the absolute convergence property holds more gener-

ically than this example. No matter how large the order 𝑚, we are still using a low order formula

(such as the midpoint formula 𝑈2) as a base. Moreover, recall that the MPF is essentially a sum of

product formulas with different numbers of time steps (for the same time interval). As the order

𝑚 increases, higher weight is given to terms in the multiproduct sum with finer meshes. Corre-

spondingly, terms which have larger time steps, and therefore may not converge properly, become

suppressed at large 𝑚. Such behavior is not reflected in our derived error bounds, so there is likely

room for improvement.

137

Practitioners in quantum simulation will likely want to know how MPFs fare against the more-

familiar and simpler Trotter techniques. To facilitate this, numerical studies across a broad range

of physically interesting systems would be desirable. Such a comprehensive analysis must be

left to future work; here we will be satisfied with comparing MPFs with Trotterization for our

spin-1/2 example. Our Trotterization is just an MPF with 𝑚 = 1, corresponding to a midpoint-

formula approximation. To facilitate as fair a comparison as possible, we will keep the number

of midpoint-formula queries between the two methods the same. That is, we will enforce the

requirement

𝑟trot = 𝑟mpf max

𝑗

|𝑘 𝑗 |

(5.129)

where 𝑟trot and 𝑟mpf are the number of time steps for Trotter and MPF, respectively. Note that the

number of midpoint queries per time step for Trotter and MPFs are 1 and max 𝑗 |𝑘 𝑗 | respectively.

Figure 5.7.1 shows the results of these head-to-head comparisons for the several values of

the magnetic field 𝐵 and rotation frequency 𝜔. The number of MPF steps 𝑟mpf is fixed at 10, a

reasonable value since it makes ΛΔ𝑡 ∼ 1 on each subinterval. As the MPF order increases, so

does the number of Trotter steps 𝑟trot by the condition (5.129). These results show that, for 𝑚 not

too large, MPFs outperform Trotterization, at a value of the error 𝜖 which is large enough to be of

practical significance for scientific or industrial applications.

Admittedly, the spin-1/2 system considered above is rather simplistic. However, we anticipate

most of the inferences drawn above to hold even as we increase the dimensionality of the Hilbert

space. For example, though the complexity of simulating 𝑈2 generally increases as dim(𝐻) grows,

it does so both for MPFs and Trotterization. Nevertheless, benchmarking of MPFs on more complex

systems would be a welcomed proof (or disproof) of concept.

5.7.2 Example 2: Spin Chain in Interaction Picture

As a first step towards more complicated many-body quantum systems, we investigate the

use of MPFs for a particular one-dimensional chain of spins with nearest-neighbor interactions.

As before, we will take advantage of a change of reference frame, allowing us to compare the

multiproduct simulations with a machine-precise simulation in an equivalent, time independent

138

Figure 5.3 Simulation error (spectral norm) of MPFs and midpoint-formula Trotterization, for the
spin-1/2 system, with number of midpoint-formula queries kept fixed between the two. Each plot
corresponds to different values for the parameters 𝐵 and 𝜔, always with 𝜃 = 𝜋/6. The number of
MPF steps 𝑟mpf is fixed at 10. The crossover point tends to occur for error 𝜖 > 10−3, which is large
enough for practical significance. Such error tolerances can be orders of magnitude larger than
those required in many quantum simulation proposals [112, 84].

frame. In pursuit of a good case study, we seek a (time independent) Hamiltonian 𝐻 = 𝐻0 + 𝐻1

which produces nontrivial time-dependence in the so-called "interaction picture." We also ask that

it satisfies a simple conservation law. A special instance of the 1D 𝑋 𝑋 model will suffice to meet

these conditions. Consider a circular chain of 𝑁 qubits with nearest-neighbor hopping interactions,

with Hamiltonian 𝐻 = 𝐻0 + 𝐻1 of the form

𝐻0 =

𝐻1 =

𝑁
∑︁

𝑘=1
𝑁
∑︁

𝑘=1

𝜔𝑘
2

𝐽𝑘
2

𝑍𝑘

(𝑋𝑘 𝑋𝑘+1 + 𝑌𝑘𝑌𝑘+1) .

(5.130)

Here, 𝜔𝑘 , 𝐽𝑘 are real, site-dependent parameters, and any index increments are done modulo 𝑁.

For any value of the parameters, the Hamiltonian conserves the total magnetization 𝜇 := (cid:205)𝑘 𝑍𝑘 .

[𝜇, 𝐻] = 0

(5.131)

Conceptually will think of 𝐻0 as a "base" Hamiltonian, with perturbation 𝐻1 generating inter-

actions, though we make no assumptions as to the smallness of 𝐻1. We will switch to an interaction

picture which is comoving with the simple dynamics of 𝐻0. In this frame, the Hamiltonian ˜𝐻 (𝑡) is

139

given by

where

˜𝐻 (𝑡) = 𝑒𝑖𝐻0𝑡 𝐻1𝑒−𝑖𝐻0𝑡
𝑁
∑︁

𝐽𝑘
2

=

𝑘=1

(𝑋𝑘 (𝑡) 𝑋𝑘+1(𝑡) + 𝑌𝑘 (𝑡)𝑌𝑘+1(𝑡))

𝑋𝑘 (𝑡) := 𝑒𝑖𝐻0𝑡 𝑋𝑘 𝑒−𝑖𝐻0𝑡 = cos(𝜔𝑘𝑡) 𝑋𝑘 − sin(𝜔𝑘𝑡)𝑌𝑘

𝑌𝑘 (𝑡) := 𝑒𝑖𝐻0𝑡𝑌𝑘 𝑒−𝑖𝐻0𝑡 = cos(𝜔𝑘𝑡)𝑌𝑘 + sin(𝜔𝑘𝑡) 𝑋𝑘

(5.132)

(5.133)

correspond to rotating the pauli vectors about the 𝑧-axis with frequency 𝜔𝑘 . We can express

equation (5.132) in terms of the time independent 𝑋𝑘 and 𝑌𝑘 of the original frame,

˜𝐻 (𝑡) =

𝑁
∑︁

𝑘=1

𝐽𝑘
2

(cid:8) cos(Δ𝜔𝑘𝑡)(𝑋𝑘 𝑋𝑘+1 + 𝑌𝑘𝑌𝑘+1) + sin(Δ𝜔𝑘𝑡) (𝑋𝑘𝑌𝑘+1 − 𝑌𝑘 𝑋𝑘+1)(cid:9),

(5.134)

where Δ𝜔𝑘 = 𝜔𝑘+1 − 𝜔𝑘 . We see that having different qubit frequencies 𝜔𝑘 on neighboring sites

should give rise to a nontrivial time-dependence in ˜𝐻. Another indication is gleaned from the

commutator of 𝐻0 and 𝐻1.

[𝐻0, 𝐻1] = −𝑖 ∑︁

𝑘

𝐽𝑘
2

(𝑋𝑘𝑌𝑘+1 − 𝑌𝑘 𝑋𝑘+1) (Δ𝜔𝑘 )

(5.135)

The time dependence in 𝐻𝐼 will be nontrivial when the commutator does not vanish, as occurs

when Δ𝜔𝑘 ≠ 0. A simple choice is to set

𝐽𝑘 = 𝐽, 𝜔𝑘 = (−1) 𝑘 𝜔.

(5.136)

That is, the qubit frequency alternates sign at each site, and the coupling is translation invariant.

For simplicity, we consider only even numbers of qubits to avoid frequency-matching at 𝑘 = 𝑁.

Plugging (5.136) into the expression for ˜𝐻 in (5.134),

where

˜𝐻 (𝑡) =

𝐽

2

(cid:0) cos(2𝜔𝑡)𝐺1 + sin(2𝜔𝑡)𝐺2

(cid:1)

𝐺1 =

𝐺2 =

𝑁
∑︁

𝑘=1
𝑁
∑︁

𝑘=1

𝑋𝑘 𝑋𝑘+1 + 𝑌𝑘𝑌𝑘+1

(−1) 𝑘 (𝑋𝑘𝑌𝑘+1 − 𝑌𝑘 𝑋𝑘+1).

140

(5.137)

(5.138)

As a final check, one can see that 𝐺1 and 𝐺2 do not commute with each other. Yet they both

commute with 𝜇. Thus, ˜𝐻 (𝑡) given in (5.137) is our model system to investigate.

Assuming ˜𝐻 commutes with an observable 𝜇, to what degree does the MPF 𝑈2,𝑚 conserve 𝜇?

Since 𝑈2,𝑚 is an algebraic combination of exponentials of ˜𝐻, 𝑈2,𝑚 also commutes with 𝜇. If 𝑈2,𝑚

were truly unitary, then the operator 𝜇 would evolve in the Heisenberg picture as

𝜇2,𝑚 (𝑡) := 𝑈†

2,𝑚 (𝑡)𝜇𝑈2,𝑚 (𝑡) = 𝜇

as it would under the exact propagator 𝑈. However, 𝑈2,𝑚 is not necessarily unitary.

𝑈†
2,𝑚 (𝑡)𝑈2,𝑚 (𝑡) ≠ 𝐼

This implies that conservation laws are only approximately conserved.

(5.139)

(5.140)

𝜇2,𝑚 (𝑡) − 𝜇 =

Because 𝑈2,𝑚 (𝑡) − 𝑈 (𝑡) ∈ 𝑂 (𝑡2𝑚+1), so is

(cid:17)

(cid:16)
𝑈†
2,𝑚 (𝑡)𝑈2,𝑚 (𝑡) − 𝐼
(cid:17)

(cid:16)
𝑈†
2,𝑚 (𝑡)𝑈2,𝑚 (𝑡) − 𝐼

.

𝜇 ≠ 0

(5.141)

Figure 5.7.2 plots the deviations in the conserved 𝜇, ∥𝜇− 𝜇2,𝑚 (𝑡) ∥, with respect to the simulation

time. As the simulation time tends to zero, we see the expected power-law scaling, as evidence

by the linear relationship on a log-log plot. For larger 𝑚, the slope and hence power 𝑝 increases,

corresponding to improved performance. We can extract the power as the slope of the line, and this

is plotted in the right frame. Notice there are sudden dips in the error at specific simulation times,

which tend to occur before reaching the power law scaling regime. This could be due to cancellation

between two terms in an error series of comparable magnitude. Similar phenomenon occurs in

several other contexts, such as the error from adiabatic evolution [135]. Conclusive identification

of these phenomenon will require further study.

Naively, we would expect 𝑝 = 2𝑚 + 1, but here we actually get slightly better: 𝑝 = 2𝑚 + 2. In

fact, this scaling can be justified. The following argument, a variant of which can be found in [28],

shows that the integrator is nearly unitary.

Theorem 5.7.1. The deviation of 𝑈2,𝑚 from being unitary obeys

∥𝑈†

2,𝑚 (𝑡)𝑈2,𝑚 (𝑡) − 𝐼 ∥ ∈ 𝑂 (𝑡2𝑚+2).

141

Figure 5.4 (left) Deviations from the conservation of magnetization 𝜇 under time-evolution by
MPFs. Note that the order 𝑚 = 1 is simply a product formula evolution, which conserves 𝜇
exactly. For small simulation times, the expected power-law scaling is observed, with larger
powers as 𝑚 increases. (right) The running power 𝑝(𝑡, 𝑡′) as defined in (5.128), with 𝑡′ = .3. Note
the plateau at 2𝑚 + 2, which indicates slightly better convergence than naively expected
(𝑝 = 2𝑚 + 1). This phenomenon generalizes to other systems and is formalized by Theorem 5.7.1.
Parameter values: 𝑁 = 4, 𝐽 = 1, 𝜔 = 4.

Proof. We suppress all function evaluations at 𝑡 when convenient. Let 𝐸 := 𝑈2,𝑚 − 𝑈, so that
𝑈2,𝑚 = 𝑈 + 𝐸. Then, using the unitarity of 𝑈 and the fact that 𝐸 ∈ 𝑂 (𝑡2𝑚+1),

where

2,𝑚𝑈2,𝑚 = 𝐼 + 𝑁 + 𝑂 (𝑡4𝑚+2)
𝑈†

𝑁 := 𝑈†𝐸 + 𝐸 †𝑈.

(5.142)

(5.143)

Since 𝑁 ∈ 𝑂 (𝑡2𝑚+1), all of its derivatives up to degree 2𝑚 vanish when evaluated at 𝑡 = 0. Hence,

it suffices to show that

𝑁 (2𝑚+1) (0) = 0.

(5.144)

We can expand this derivative in terms of 𝐸 and 𝑈 using the binomial theorem. When we evaluate

at 𝑡 = 0, those terms with derivative less than degree 2𝑚 + 1 in 𝐸 vanish. We are left with

𝑁 (2𝑚+1) (0) = 𝐸 †(2𝑚+1) (0)𝑈 (0) + 𝑈†(0)𝐸 (2𝑚+1) (0).

(5.145)

We have 𝑈 (0) = 𝑈†(0) = 𝐼. Moreover, by the time-symmetric property of 𝑈 and 𝑈2,𝑚, 𝐸 (𝑡) is also

symmetric. Therefore

𝐸 †(2𝑚+1) (0) = 𝐸 (2𝑚+1) (−𝑡)

(cid:12)
(cid:12)
(cid:12)𝑡=0

= −𝐸 (2𝑚+1) (0).

(5.146)

142

Hence, the two terms in (5.145) cancel, yielding 𝑁 (2𝑚+1) (0) = 0. This completes the proof.

□

In summary, though MPFs do not inherently preserve commutations laws, the error is due to

nonunitarity in 𝑈2,𝑚. This can be bounded and reduced in a systematic way, either by decreasing

the time step or increasing the MPF order.

5.8 Discussion

In this chapter, we presented an algorithm for time dependent Hamiltonian simulation that

uses multiproduct formulas to boost the accuracy compared to product formula simulation. Our

algorithm inherits the commutator scaling of product formulas, giving a benefit over comparable

methods such as the Dyson series approach. We provide a rigorous characterization of the sim-

ulation error as well as query computational complexity. Numerical demonstrations validate the

effectiveness of time dependent MPFs in achieving high-accuracy simulations.

Several avenues for future research are immediately apparent from this chapter. First, a proof

of Conjecture 1 is highly desirable. Currently, we are investigating modifications of the clock

space construction that keep the clock state width 𝜎 fixed, allowing for completion of the argument.

Numerical demonstrations beyond the simple examples here would be desirable for showing scaling

to larger systems, and of course, eventually the method should be tested on actual quantum hardware.

To summarize, time dependent MPF simulation is a new algorithm which complements many

existing approaches and likely will perform well on systems with a large degree of locality.

5.9 Algorithm for Time Mesh

For completeness, I include the greedy algorithm for generating the time mesh used in the MPF

simulation. I thank Alessandro Roggero for devising the approach presented below.

The mesh construction of Section 5.5, although theoretically sound, is not directly imple-

mentable since it requires knowing the total number of steps while constructing each new point

based on local data. To avoid this issue, as well as the restriction | (cid:164)Λ(𝜏)| ≤ 𝐾Λ2(𝜏) we seek a

simple-to-use greedy algorithm.

One possibility is to use a direct approach which first selects a candidate number of steps 𝑟try.

Starting from 𝑟try = 1, we then build recursively a sequence of times using the condition (see

143

Eq. (5.83) in the main text)

max
𝑡∈[𝑡𝑖−1,𝑡𝑖]

Λ2𝑚 (𝑡) (𝑡𝑖 − 𝑡𝑖−1) ≤

1
41

(cid:18)

𝜖
0.32∥𝑎∥1𝑟

(cid:19) 1/(2𝑚+1)

,

(5.147)

with 𝑟 = 𝑟try. Starting from 𝑡0 = 0 and looking for the largest 𝑡𝑖 that satisfies the condition, we finally

check whether the generated number of intervals is greater than 𝑟try in which case we increase 𝑟try

by one and repeat. When the algorithm stops at the optimal value 𝑟opt, we have performed a total

of 𝑟opt(𝑟opt + 1)/2 non-linear optimization steps, each one requiring multiple evaluations of the left

hand side of Eq. (5.147). This can be very demanding when the left hand side of Eq. (5.147) is

expensive to evaluate and the optimal number of intervals is around half the upperbound

(cid:18)

𝑟max =

41(𝑡 − 𝑡0) max
𝜏∈[𝑡0,𝑡]

Λ2𝑚 (𝜏)

(cid:19) 2𝑚+1

2𝑚 (cid:18) 0.32∥𝑎∥1

(cid:19) 1
2𝑚

𝜖

(5.148)

obtained considering identical intervals and bounding Λ2𝑚 (𝑡) with its maximum value over the

whole simulation interval [0, 𝑇]. In this case, finding an approximation to the optimal decomposi-

tion requires 𝑂 (𝑟 2

max) optimization steps, each one requiring multiple evaluations of the lefty hand

side of Eq. (5.147).

We now describe an alternative approach which determines 𝑟opt within a factor of 2 and uses

only 𝑟max evaluations of max𝑡∈[𝑡𝑖−1,𝑡𝑖] Λ2𝑚 (𝑡) and additional 𝑂 (log(𝑟max)𝑟max) simple arithmetic

operations. This procedure can be used to find a viable, and approximately optimal, decomposition

of the time interval or as a good starting point to find the optimal one using a procedure as the one

described above. The idea is to start by decomposing the interval [0, 𝑇] into 𝑟max segments with

equal length and storing the maximum of Λ2𝑚 (𝑡) in each segment in an array 𝐴 of size 𝑟max. We

then introduce an additional array of the same size

𝐿𝑚 =

(cid:20)

max
𝑘 ≤𝑚

(cid:21)

𝑚

𝐴𝑘

𝑇
𝑟max

,

together with an additional set of vectors of the same size

(cid:20)

𝑅(𝑛)

𝑚 =

max
𝑛≥𝑘 >𝑚

(cid:21)

𝐴𝑘

(𝑛 − 𝑚)

𝑇
𝑟max

,

(5.149)

(5.150)

with 𝑛 an additional index between 1 and 𝑟𝑚𝑎𝑥. The first vector stores the left hand side of

Eq. (5.147) for the interval up to the 𝑚-th time while the second vector stores the same information

144

for the interval starting at the 𝑚-th time and ending at the 𝑛-th one. The algorithm proceeds by

splitting the time interval recursively into two parts so that the left hand side of Eq. (5.147) takes

(approximately) the same value on both halves (ie. we are splitting the error equally on both sides).

At every iteration the number of intervals doubles and the right hand side of Eq. (5.147) shrinks

accordingly. We stop the procedure once Eq. (5.147) is satisfied on one interval (since we are

guaranteed it will in all others). The procedure will stop at some 𝑟𝐾 at which point we know the

optimal value 𝑟opt is in [⌈𝑟𝐾/2⌉, 𝑟𝐾]. The algorithm can then be described as follows

1. Compute 𝐿𝑚 for all 𝑚 = 1, ..., 𝑟max

2. Set 𝑛 = 𝑟max and 𝑟 = 2

3. Compute the elements of 𝑅(𝑛)

𝑚 for all 𝑚 = 1, ..., 𝑛 − 1

4. Initialize an auxiliary array 𝐷𝑚 as 𝐷𝑚 = 𝐿𝑚 − 𝑅(𝑛)
𝑚

5. Find the least index 𝑘 for which 𝐷 𝑘 > 0

6. If 𝐿 𝑘 is less than the right hand side of Eq. (5.147) with the current value of 𝑟, set 𝑟𝐾 = 𝑟 and

exit

7. If 2𝑟 ≥ 𝑟max set 𝑟𝐾 = 𝑟max and exit

8. set 𝑟 = 2𝑟, 𝑛 = 𝑘 and repeat from step 3

Step 1 requires 𝑟max operations while Steps 3 and 4 cost 𝑛 operations each. Since the number
of iterations is bounded by log2(𝑟max), their combined cost is bounded by 2 log2(𝑟max)𝑟max. If we
use binary search, Step 5 costs log2(𝑛) operations so its total cost is at most log2(𝑟max)2 operations.

From this analysis we see that Steps 3 and 4 are the most expensive ones and they dominate the

cost of the scheme. On exit we have 𝑟𝐾 ≈ 𝑟opt together with the first interval [𝑡0, 𝑡1]. The rest of

the intervals can then be found keeping 𝑟 = 𝑟𝐾 fixed with additional 𝑂 (𝑟max) operations.

145

CHAPTER 6

A SIMULATION MIXED BAG

Each of the prior chapters of this dissertation, starting with Chapter 3, focuses on a (mostly)

self-contained theoretical project. However, I also contributed to several other projects during

my PhD, especially during earlier years. These projects are more concerned with applications on

noisy, small-scale devices than with methods suited for relatively noiseless processors. While my

contributions were often of a mathematical and theoretical flavor, the ultimate goal was to apply our

algorithmic gadgets to existing quantum devices, emphasizing demonstration over rigorous proof.

Each section of the present chapter corresponds to one of these projects, presented in the order

they were completed, with corresponding publication referenced within the first few paragraphs.

I will begin with a topic that technically falls outside of quantum computing proper, in which we

propose investigating discrete scale invariance and anomalous symmetry breaking with trapped-

ion quantum simulators. Moving back to quantum computing, I will present a heuristic quantum

algorithm for preparing low-lying energy states known as the Projected Cooling algorithm. Finally,

I will discuss the aptly but oddly named Rodeo Algorithm, which is a randomized iterative phase

estimation algorithm for determining eigenvalues and preparing eigenstates of an observable,

provided that observable can be efficiently time evolved.

6.1 Discrete Scale Invariance on Trapped-Ion Systems

Closely related to the idea of simulation by quantum computation is the emulation of a desired

Hamiltonian on a controllable and accessible quantum system, namely, analog Hamiltonian simu-

lation. See Subsection 2.7.4 for a short background and references to more detailed introductions.

In one of my first projects as a PhD student, I proposed, along with collaborators [82], a scheme

for simulating scale invariant Hamiltonians on trapped-ion quantum simulators. As of writing,

trapped ions are one of the most mature platforms for exquisite manipulation of quantum systems.

We begin this topic by introducing some of the deep physics ideas motivating our study, such as

anomalous symmetry breaking. Then we will discuss our proposal for investigating these concepts

on certain Hamiltonians accessible to trapped-ion simulators. We will conclude with some of my

146

findings on the self-similar dynamics which these systems exhibit in for certain nearly-unbound

states.

6.1.1 Scale Invariance and Quantum Anomalies

Understanding how a system changes with scale is a fundamental question in physics. For

example: How do aspects of a microscopic system affect behavior at larger scales? Which of these

aspects are relevant, and which get washed out by averaging or some other mechanism? In modern

parlance, such questions are formalized in the theory of renormalization [91], which touches on

essentially all aspects of modern physics, from the Standard Model to phase transitions. Closely

related is the concept of emergent phenomena, which deals with how complex many-body systems

achieve their properties, not through microscopic properties themselves, but through the complex

interactions of these properties.

It is remarkable that the new descriptions produced by these

microscopic interactions are even intelligible. Renormalization provides a partial answer, showing

how the effect of changing scales can often be incorporated, to good approximation, by changes to

couplings in the physical theory.

Another important concept in physics is symmetry. Symmetries provide such a powerful

framework for understanding a physical system that little progress in physics could be made

without them. Noether’s theorem relates certain continuous symmetries to conservation laws,

and in practice symmetries allow for the reduction of computation and analysis needed to solve a

problem. While symmetries are by nature elegant and simplifying, the breaking of symmetry has

proven an essential idea to modern physics. In this project, we explored how quantum anomalies–the

breaking of symmetry in a classical Hamiltonian by quantization–can be studied using a class of

Hamiltonians that are implementable on existing trapped-ion simulators.

Symmetry is merely a change that, in fact, leads to no change at all. When a scale transformation

is performed, yet the system looks the same at this new scale, we have scale invariance. This is

an important idea in field theory and renormalization (fixed points and conformal field theory).

Even more "mundane" standard quantum mechanical systems can exhibit scale invariance. Take,

147

for example, the 1/𝑟 2 potential

𝐻 = 𝛼

𝑝2

2

+

𝛽
𝑟 2

(6.1)

for 𝛼 ∈ R+ and 𝛽 ∈ R. Performing a canonical scale transformation 𝑟 ↦→ 𝜆𝑟 and 𝑝 ↦→ 𝜆−1 𝑝 for

any 𝜆 ∈ R+, we find that 𝐻 (𝜆𝑟, 𝑝/𝜆) = 𝜆−2𝐻 (𝑟, 𝑝). Classically, this leads to a trivial change in the

phase space dynamics whereby the trajectories are the same, but time rescales as 𝜆2𝑡. Quantum

mechanically, the time evolution operator is rescaled by the same factor. Remarkably, even this

simple example leads to a quantum anomaly for sufficiently attractive potentials [20, 62]. The full

scale invariance is broken to a discrete one, giving rise to a tower of bound states approaching

𝐸 → 0 related by a geometric sequence. Thus, this simple Hamiltonian, which may arise from the

interaction of an electromagnetic dipole and point charge, already exhibits a surprising richness.

What’s more, the potential turns out to be intimately related to the Efimov effect [41, 100], a curious

phenomenon in which bosons, with short range interactions, cannot form a two-body bound state,

but three bosons together have an infinite number of bound states. The binding of three particles,

but not two, recalls the intriguing Borromean rings [34], which are linked such that cutting any

single ring unlinks all the rings. Since the original discovery, Efimov physics has been linked to a

broad class of phenomena, and theoretical interest has been recently growing [100].

6.1.2 Trapped-Ion Systems

One of the most developed hardware platforms for quantum information processing is the ion

trap [19], first proposed as a platform for universal quantum computation shortly after Shor’s

groundbreaking factoring algorithm [30]. Atomic ions, typically Ytterbium or one of several

alkaline earth metals, are trapped using lasers in a 1D chain. External control and readout of the

system is mediated by laser pulses, and interactions between the ions are mediated, interestingly,

through coupling with phononic modes [101] in the 1D chain. Trapped ions possess a number

of internal states that can be used as the informational degree of freedom (e.g., qubit) including

hyperfine, Zeeman, optimal, and fine transitions. The impressive degree of control, all-to-all

connectivity, high fidelity of operations, and long coherence times make trapped ions a promising

platform for quantum hardware experiments and computations.

148

Although the technology is being pursued for general computation, these systems can also be

operated as analog simulators. Our study is motivated by a particular long range interaction that

can be engineered between the ions. Using two hyperfine "clock" states, the following effective

spin Hamiltonian can be approximately generated [104].

∑︁

∑︁

𝐻 =

𝑖, 𝑗

𝑘=𝑥,𝑦,𝑧

𝑖 𝑗 𝜎 (𝑖)
𝐽 𝑘

𝑘 𝜎 ( 𝑗)

𝑘 +

∑︁

∑︁

𝑖

𝑘=𝑥,𝑦,𝑧

𝐵𝑖𝑘 𝜎 (𝑖)
𝑘

(6.2)

Here 𝜎 (𝑖)
𝑘

is the 𝑘th Pauli matrix on site for ion 𝑖. The parameters 𝐽 are related to the spatial

coordinates of the ions and can be tuned separately for each 𝑘 using laser pulses. At long ranges,

the couplings fall off as

𝐽 𝑘
𝑖 𝑗 ≈

𝐽 𝑘
0
|𝑖 − 𝑗 |𝛼

(6.3)

for chosen 𝛼 ∈ (0, 3) [104, 72] and couplings 𝐽 𝑘
0 .

It is the long-range character (6.3) that we find particularly interesting. By factorizing (6.2) into

and 𝑥𝑦 piece and a 𝑧 piece, and choosing appropriate coupling parameters, we can created effective

kinetic-potential Hamiltonians with scale invariant properties. First, making the choice

𝑖 𝑗 = 𝐽 𝑦
𝐽𝑥

𝑖 𝑗 ≡ 𝐽𝑖 𝑗 ,

𝐽 𝑧
𝑖 𝑗 = 𝑉𝑖 𝑗 ,

𝐵𝑖𝑘 = 𝛿𝑘 𝑧𝑈𝑖

(6.4)

we find that the Hamiltonian factorizes as 𝐻 = 𝑇 + 𝑉 + 𝑈, where 𝑇 is a kinetic term, 𝑉 is a potential

interaction term, and 𝑈 is a on-site potential. With this parametrization, the full Hamiltonian

becomes

𝐻 =

∑︁

𝑖≠ 𝑗

𝐽𝑖 𝑗 (𝑋𝑖 𝑋 𝑗 + 𝑌𝑖𝑌𝑗 ) +

∑︁

𝑖≠ 𝑗

𝑉𝑖 𝑗 𝑍𝑖 𝑍 𝑗 +

∑︁

𝑈𝑖 𝑍𝑖.

𝑖

(6.5)

The "particles" of the system are given by the spin state: we say that there is a particle at site 𝑗

if a 𝑍 𝑗 measurement yields +1, i.e., we equate particle occupancy with the binary value of the 𝑍 𝑗

observable. Notice that (cid:205)𝑖 𝑍𝑖 is conserved by 𝐻, hence particle number is conserved. We can

interpret these particles as hard-core bosons, with raising and lowering operators given by 𝜎±, and

strong repellant interactions at short distances preventing multiple occupancy of a site. Hard-core

bosons might serve as a reasonable model for a collection of nuclei with even nucleon number,

since nuclei are repulsive at low energies.

149

We want to consider the Hamiltonian (6.5) as a discretization of some continuous-variable

system with scale invariance. As such, we will need to consider low energies where wavelengths

are large compared to the lattice. Considering only the kinetic term 𝑇 alone, with asymptotic

form (6.3) of the hopping coefficient, and neglecting boundary effects by assuming a large chain,

the eigenenergies are given by

𝐸 ( 𝑝) = 4𝐽0

∞
∑︁

𝑚=1

1
𝑚𝛼 cos( 𝑝𝑚).

Performing a low momentum expansion, we find that for 𝛼 < 3

𝐸 ( 𝑝) = 2𝐽0 sin(𝛼𝜋/2)Γ(1 − 𝛼)| 𝑝|𝛼−1 + 𝑂 ( 𝑝2).

(6.6)

(6.7)

Here is the first part of our scale invariant Hamiltonian. Now we consider the potential part. We

fix one of our bosons ("spin up") on one of the sites, say site 0, with a very deep on-site potential

𝑈0. We then add a second boson and tune the potential interaction 𝑉𝑖 𝑗 such that, at large distances,

𝑉𝑖 𝑗 ∼

𝑉0
|𝑖 − 𝑗 |𝛼−1

.

Then, we see that the total Hamiltonian

𝐻 = 2𝐽0 sin(𝛼𝜋/2)Γ(1 − 𝛼)| 𝑝|𝛼−1 +

𝑉0
|𝑟 |𝛼−1

+ 𝑂 ( 𝑝2)

(6.8)

(6.9)

is scale invariant at low momenta. With choice of parameters 𝐽0 < 0, 𝑉0 < 0, this Hamiltonian

has an attractive potential. In our paper’s Supplemental Material [82], we provide the limit cycle

boundary 𝐸 = 0 state as well as the discrete scaling factor relating the bound state energies near

this threshold.

6.1.3 Self-Similar Dynamics

In the previous sections, we have discussed discrete scale invariance in the bound state spectra

of certain Hamiltonians, and how these could be constructed on trapped-ion systems. Discrete scale

invariance implies self-similarity as one changes their "field of view" by certain discrete amounts.

Besides providing equilibrium information via the Gibbs state, the bound state spectrum impacts

the closed-system dynamics. Can the self-similarity of the bound state spectrum manifest in the

150

dynamical evolution of a chosen initial state? We will show that the answer is yes, though the initial

state must be chosen carefully and may be difficult to prepare.

The idea is to prepare a weakly bound state that overlaps many of the eigenstates just below the

𝐸 = 0 threshold. Take as initial state

|𝜓(0)⟩ =

∞
∑︁

𝑛=0

𝑐𝑛 |𝑛⟩

(6.10)

where 𝑛 indexes the bound states of energy 𝐸𝑛 < 0, with some chosen 𝐸0 as the lowest energy state.
These energies are related in geometric sequence as 𝐸𝑛 = 𝐸0/𝜆𝑛 for some 𝜆 > 1. The time evolved

state is given by

|𝜓(𝑡)⟩ =

∞
∑︁

𝑛=0

𝑐𝑛𝑒−𝑖𝐸𝑛𝑡 |𝑛⟩ =

∞
∑︁

𝑛=0

𝑐𝑛𝑒−𝑖𝐸0𝑡/𝜆𝑛

|𝑛⟩ .

(6.11)

Suppose now we rescale 𝑡 by 𝜆. Because 𝐸𝑛−1 = 𝐸𝑛𝜆, this is simply (cid:205)𝑛 𝑐𝑛𝑒−𝑖𝐸𝑛−1𝑡 |𝑛⟩. Without
relating the coefficients 𝑐𝑛, there isn’t much more that can be said.

Imposing the condition

𝑐𝑛 = 𝛾𝑐𝑛−1, however, we find

|𝜓(𝜆𝑡)⟩ = 𝛾

∞
∑︁

𝑛=1

𝑐𝑛−1𝑒−𝑖𝐸𝑛−1𝑡 |𝑛⟩ + 𝑐0𝑒−𝑖𝐸0𝜆𝑡 |0⟩

= 𝛾𝑈+ |𝜓(𝑡)⟩ + 𝑐0𝑒−𝑖𝐸0𝜆𝑡 |0⟩

(6.12)

initial state |𝜓(0)⟩ on the subspace span{|𝑛⟩}∞

where 𝑈+ |𝑛⟩ := |𝑛 + 1⟩. Besides the piece proportional to |0⟩, we have a representation of the
𝑛=1. The scaling factor 𝛾 ∈ C must satisfy |𝛾| < 1
for normalization purposes, and in fact |𝑐0|2 + |𝛾|2 = 1. Already, we see some manifestation of

self-similarity.

Some feasible measurement must be performed to extract the information about the state (6.12).

Let’s consider the overlap observable 𝑊 (𝑡) := ⟨𝜓(0)|𝜓(𝑡)⟩, which might be obtained with two

copies of the initial state, one of which time-evolved, then performing some version of the SWAP

test. We find

𝑊 (𝑡) = (1 − |𝛾|2)

|𝛾|2𝑛𝑒−𝑖𝐸0𝑡/𝜆𝑛 .

∞
∑︁

𝑛=0

(6.13)

151

Up to normalization and nomenclature, the real part of this is nothing more than a Weierstrass

function, defined in the original paper as

∞
∑︁

𝑛=0

𝑎𝑛 cos(𝑏𝑛𝜋𝑥).

In particular, we make the identifications

𝑎 = |𝛾|2,

𝑏 = 𝜆−1,

𝑥 =

𝐸0𝑡
𝜋

.

(6.14)

(6.15)

For certain values of the parameters 𝑎, 𝑏 the function (6.14) is a fractal, being continuous everywhere

but differentiable nowhere. This is not the case we are currently in; our evolution is mathematically

smooth. As 𝑛 increases in the series, the frequencies 𝑏𝑛 decrease exponentially, leading to an utterly

smooth behavior. This is not surprising, since all of the bound eigenstates states present in our

initial state have frequencies within [𝐸0, 0].

Despite this, we can recover an approximate self similarity at long time scales. Consider

𝛾 = 1 − 𝛿, with 𝛿 > 0 taken very small. Then, for values 𝑁 ∈ Z+ such that 𝑛𝛿 ≪ 1 for all 𝑛 < 𝑁, the

first 𝑁 levels have approximately equal weight in the superposition. Shifting 𝑡 ↦→ 𝜆𝑡 then leaves the

function approximately unchanged, up to a small high-frequency component. For these 𝑁 levels,

then, we get

𝑁
∑︁

𝑛=0

cos(𝜆−𝑛𝐸0𝑡)

(6.16)

exhibiting self similar behavior. Moreover, if we rescale to 𝑡 ↦→ 𝜆𝑛𝑡 and reindex 𝑛 ↦→ 𝑁 − 𝑛, we

get something which looks like a truncation of a Weierstrass function

𝑁
∑︁

𝑛=0

cos(𝜆𝑛𝐸0𝑡).

(6.17)

In short, by zooming out to larger time scales, the short time scales appear fractal. The Weierstrass

function is famous as being an example of a continuous function that is differentiable nowhere. It

is also a fractal, with fractal dimension

𝐷 = 2 +

log 𝑎
log 𝑏

(cid:18)

1 −

= 2

(cid:19)

.

log|𝛾|
log 𝜆

(6.18)

152

In our case 𝐷 = 2, so our curve is space filling. Different choices of coefficient 𝑐 𝑗 , in which lower

frequencies have higher amplitude, could produce different values of 𝑎 and therefore different

fractal dimensions.

6.1.4 Discussion

In this Section, we show how trapped-ion quantum simulators can be used to investigate

Hamiltonians with anomalous symmetry breaking due to quantization, with applications to Efimov

physics. We characterized the nature of the discrete scale invariance of the bound state spectrum

for a family of scale-invariant Hamiltonians. Finally, we indicate how self-similar dynamics,

reminiscent of the Weierstrass function, can be obtained through particular state preparation and

measurement.

Unfortunately, technological challenges remain to implementing the low energy Hamilto-

nian (6.9), particularly due to the need for long wavelengths relative to the ion spacing. As of

writing, trapped ion quantum computers are of size at most 60 [50, 140], which appears small

enough to introduce unwanted boundary effects. While there are no fundamental limitations to the

size of the ion trap, implementing interactions between the ions becomes increasingly challenging

and expensive as the size scales up [98]. We hope these challenges may be resolved by future

improvements in technology or clever implementation of our approach.

6.2 Projected Cooling Algorithm

Having considered analog quantum simulators, we return back to digital quantum computing,

but still with an analog flavor. Absent of error correction and logical qubits, current quantum

computers mainly reside in the regime of Noisy Intermediate Scale Quantum (NISQ) devices.

Such devices implement imperfect operations and are subject to decoherence, severely restricting

maximum computation time. For algorithm developers looking to find near-term application of

quantum computers, it is important that the proposed algorithms are relatively noise insensitive.

In this section, we will discuss the Projected Cooling Algorithm [83], which prepares ground

states, or generally low-lying states, of a kinetic-potential Hamiltonian with localized interactions

and translation-invariant kinetic terms. Such systems are common in nuclear physics, as nuclear

153

interactions are spatially local and often only a few deeply bound states exist. I will discuss the

main idea of the algorithm, followed by an application to the Dirac delta potential.

6.2.1 Background

Preparing ground states of a Hamiltonian is valuable for both scientific computing and mathe-

matical optimization. This is known to be a hard problem in general [78] and even determining basic

properties, such as a the existence of a spectral gap, is undecidable [35]. However, the importance

of determining ground state properties is so great that generic hardness is not a deterrent to trying.

Moreover, generic hardness of the ground state problem does not necessarily imply hardness among

the instances of "physical interest."

Several heuristic or partial algorithms exist for preparing ground states, or low-lying eigenstates.

One of these is the adiabatic algorithm, in which a ground state of a simple Hamiltonian 𝐻0 is

prepared, then evolved according to a slowly varying time dependent 𝐻 (𝑡) which takes 𝐻0 to the

Hamiltonian 𝐻 of interest. The premise of the algorithm, as the name suggests, rests on adiabaticity:

provided that the evolution is "sufficiently slow," the state |𝜓(𝑡)⟩ will remain in the ground state of

𝐻 (𝑡) for any time 𝑡. The slowness required is related to the inverse gap between the ground state

and the next excited state at any given time. Unfortunately, it can be challenging to ensure that the

gap does not decrease exponentially with system size, despite clever choice of initial Hamiltonian

and trajectory 𝐻 (𝑡) in model space. As another example, we briefly mention Variational Quantum

Eigensolver (VQE) [22], which is an optimization approach to finding the ground state whereby

circuit parameters are optimized to reduce the energy of the state. Unfortunately, variational

methods are known to suffer from barren plateaus [95], which provide a poor optimization landscape

for gradient-based methods to succeed.

A distinct class of approaches, in which we include projected cooling, may be characterized as

measurement-based. Simple projective measurements, as the name suggests, project onto a state,

or subspace, determined by the measurement outcome. By performing a measurement compatible

with the energy basis, it is possible to post-select on measuring a result compatible with the ground

state. The main limitation arises when the probability of successful measurement is vanishingly

154

small. Indeed, for a randomly prepared initial state, the expected overlap with the ground state

should decrease exponentially with system size. A natural objection to this grim outlook is the

existence of clever ansatzes, such as Hartree-Fock states, that have been developed in the domain

sciences long before quantum computing, which can be expected to have much higher overlap than

a randomly chosen state.

The canonical and highly general measurement-based scheme for state preparation is the suite

of phase estimation algorithms, to be discussed in greater detail in the next Section. At a high

level, phase estimation is nothing more than a projective measurement in the eigenbasis of a

chosen unitary 𝑈. When 𝑈 is a time-evolution operator for time independent 𝐻, this corresponds

to a measurement of energy states. Given initial state |𝜓0⟩ with fidelity 𝑝gs with respect to the

ground space of 𝐻, a phase estimation protocol will produce the (approximate) ground state with

(approximate) probability 𝑝gs. Thus, repeating 𝑂 (1/𝑝gs) times is sufficient to produce the ground

state.

The generality of phase estimation suggests a tradeoff in the form of computational difficulty.

Phase estimation algorithms require at least one auxiliary qubit to use as a control register for a

controlled time evolution. Moreover these operations do not necessarily respect hardware connec-

tivity, as the control register must talk to all qubits in the main system. For present hardware, these

demands can be prohibitive.

6.2.2 Projected Cooling

To avoid the demands imposed by phase estimation, focusing on a more specific class of

Hamiltonians is desirable. We take inspiration from nuclear physics. Nucleons exhibit strong,

short-range interactions, and thus their bound states are also spatially localized. Many nuclei have

only a few deeply bound states, and possibly many more shallow bound states. For example, the

simple deuteron is know to have only one bound state. When nucleons are freed from the short-

range potential, they enter scattering, or continuum states that evolve according to a translationally-

invariant kinetic Hamiltonian. These scattering states tend to disperse away from the interaction

region, while the bound states naturally remain localized in the interaction region.

155

The above discussion suggests a simple criterion for distinguishing bound and unbound states

for such systems: long-term localization. Evolved under the nuclear Hamiltonian, bound states will

stay localized in the interaction region, while unbound states will disperse away. A measurement

of the nuclear configuration (i.e., many-particle position measurement) should distinguish the two

cases. To retain coherence, a binary measurement should be made which only records whether

particles are found outside the range of interaction. Assuming none are, we can expect the system

to have lower energy content, since the particle distance from the interaction region is correlated

with energy.

Such reasoning is the basis of the Projected Cooling Algorithm. We begin by preparing an

initial state localized with respect to the interaction. We then perform a time-evolution according to

the natural Hamiltonian of the system for some time 𝑇. Finally, a binary measurement is performed

which asks whether or not particles are found outside the region in which the potential 𝑉 has

greatest support.

It is reasonable to expect the initial state to have higher overlap with the ground state, barring

special symmetry, than a generic state far from the interaction region. This suggests simple yet

effective ansatzes are available for these localized systems. The simulation time 𝑇 required depends,

in principle, on the nature of the potential. Resonances will tend to weaken the effectiveness of the

algorithm by keeping continuum states around longer. At the same time, 𝑇 cannot be chosen longer

than the velocity of the dispersive component of the wavefunction, or else there will be reflections

due to the finite box size. The final measurement 𝑀 is in the configuration basis, which can be made

simple by choosing it as our computational basis. Altogether, no auxiliary registers and additional

controlled operations are necessary. The time evolution itself can be expected, generally, to be the

most difficult subroutine of the algorithm.

6.2.3 Application to Approximate 1D Dirac Delta Potential Well

We demonstrate the Projected Cooling procedure with a 1D particle in an attractive, localized,

deep potential square well. The Hamiltonian is

𝐻 =

𝑝2
2𝑚

+ 𝑉

156

(6.19)

where

𝑉 (𝑥) =

−𝑉0

|𝑥| < 𝐿

0

|𝑥| ≥ 0

(6.20)





for 𝑉0, 𝐿 > 0. We take a discretization of this into 2𝑁 sites labeled 𝑛 = −𝑁, . . . , 𝑁. We aren’t

concerned so much with the interior details of the potential as much as its localization, for this

simple example. As such, we will assume only the 𝑛 = 0 point is located within the potential; that

is, the lattice spacing 𝑎 is larger than 𝐿. The discretized potential then becomes simply

ˆ𝑉 = −𝑉0|0⟩⟨0|.

For the kinetic term, we take a symmetric finite difference approximation. That is,

ˆ𝐾 =

1
2𝑚𝑎2

(2𝐼 − 𝑈+ − 𝑈−)

(6.21)

(6.22)

where 𝑈+ = 𝑈†

− is the right-shifting unitary operation 𝑈+ |𝑛⟩ = 𝑈+ |𝑛 + 1⟩. The discretized
Hamiltonian ˆ𝐻 = ˆ𝐾 + ˆ𝑉 has a bound state spectrum that can be analyzed using an ansatz borrowed

from the continuum Dirac delta potential. Neglecting finite boundary, we take an ansatz bound

state of the form

|𝜅⟩ =

∞
∑︁

𝑛=−∞

𝑒−𝜅𝑎|𝑛| |𝑛⟩ .

(6.23)

Applying ˆ𝐻 to |𝜅⟩, we see that |𝜅⟩ is an eigenstate with energy 𝐸 provided the following two

conditions are satisfied.

𝐸𝑚𝑎2 = 1 − cosh(𝜅𝑎)

1 − 𝑒−𝜅𝑎 = 𝑚𝑎2(𝑉0 + 𝐸)

(6.24)

This transcendental equation admits a unique solution for 𝜅 > 0 and 𝐸 < 0. Thus, provided our

cutoff 𝑁 is sufficiently large, we might expect to find such a state following Projected Cooling.

For our numerical example, we work in lattice units where 𝑎 = 1, and we will also take

𝑚 = 𝑉0 = 1. The full discretized Hamiltonian is then

ˆ𝐻 = (𝐼 −

1
2

𝑈+ −

1
2

𝑈−) − |0⟩⟨0|.

(6.25)

157

We now need to discuss a mapping of this Hamiltonian onto a set of qubits. One natural choice

is to encode position into the computational basis. This has the advantage of requiring 𝑂 (log 𝑁)

qubits representing the system, which is valuable because Projected Cooling requires a large enough

region for the unbound states to disperse into. In contrast, we could directly represent each lattice

site with a qubit. This leads to a less-favorable 𝑂 (𝑁) scaling, but the advantage is that multiple

particles could be allowed, with hard-core repulsion preventing multiple occupancy. Moreover, the

operations 𝑈+, 𝑈− are much simpler to implement in this "unary" encoding.

The case of unary encoding is discussed in the paper (Model 1A) [83]. Here, we supplement this

work with a discussion of the binary encoding approach. The number of qubits needed to represent

the system scales as 𝑛 ∈ 𝑂 (log 𝑁). Preparing Gaussian wavepackets on a quantum register is a

well-studied problem [110, 80, 79], and we assume this can be done efficiently.

For simulating ˆ𝐻, several methods could be employed. Trotterizing along the two terms ˆ𝐾 and

ˆ𝑉, one could simulate ˆ𝑉 using a 𝐶𝑛−1(𝑅𝑧) gate, and ˆ𝐾 by diagonalizing via the Quantum Fourier

Transform. Alternatively, we observe that ˆ𝐻 can be expressed as a linear combination of unitaries

(LCU), hence is amenable to simulation by qubitization. The number of required queries to the

block-encoding "select" SEL and "prepare" PREP circuits scales as 𝑂 (𝑇 + log 1/𝜖) for simulation

time 𝑇 and accuracy 𝜖. The PREP circuit is only on two qubits, because once the identity terms are

removed, there are only 3 unitary pieces. The SEL cost is dominated by the controlled incrementer,

which requires 𝑂 (𝑛2) = 𝑂 (log2 𝑁) CNOT gates [85]. The potential part of SEL requires the
reflection operator 𝐼 − 2 (cid:12)

(cid:12), controlled on the "prepare register." This requires only 𝑂 (𝑛)

(cid:12)0⊗𝑛(cid:11) (cid:10)0⊗𝑛(cid:12)

gates. The number of auxiliary qubits for the LCU is 2, and does not change with system size.

Once the time evolution is performed, a measurement must be performed to determine if the

particle is found significantly outside the region of interaction. A simple computational basis (i.e.,

position) measurement won’t do, since this will destroy the state of interest.

instead, a binary

measurement must be done which asks whether the position of the particle is outside of a range 𝑅

determined by the locality of the state. This can be done with a comparator circuit [115] which

requires 𝑂 (𝑛) gates and a couple of auxiliary qubits to be measured.

158

6.2.4 Discussion

Here we considered the Projected Cooling algorithm for preparing ground states using the

dispersion of unbound states away from the interaction region. While we motivated our approach

using nuclear systems, the method can be applied to any system with attractive, localized potential

interactions and a translation-invariant kinetic energy. Following our initial work, the method

was used to investigate the transverse Ising model [58], with success for models which exhibited

dispersion rather than localization. Like other measurement-based state preparation algorithms,

success is contingent upon having sufficient overlap with the bound state of interest. For localized

interactions, an effective ansatz may correspond to, for example, a Gaussian packet of width on the

order of the interaction length.

Determining the required 𝑇, 𝜖 and 𝑁 analytically to ensure good fidelity with the ground state

is beyond our scope. In our paper, we mainly employ classical numerical simulations that suggest

good convergence to the desired state, and refer the reader to these results [83]. However, a more

careful theoretical treatment is left to be desired. This might be done using a scattering theory

treatment. This would likely elucidate the role of resonances, which should propagate slowly and

thus hinder the method’s effectiveness.

6.3 Rodeo Algorithm

This final section of the chapter concerns a new addition to the suite of iterative Quantum Phase

Estimation (IQPE) protocols known as the Rodeo Algorithm. The original algorithm was introduced

in [29], where it was tested on the Heisenberg model using classical simulations. Subsequently, an

actual quantum computation on IBM quantum computer Casablanca was performed for a simple

single-qubit Hamiltonian [106]. In more recent work [11], the Rodeo Algorithm was tested on two-

qubit Hamiltonians and used to benchmark a protocol for efficiently compiling complex sequences

of controlled operations. Rather than cover all of these works in detail, here I will describe the

principles of the algorithm, its relation to iterative phase estimation, and my contributions to the

theoretical characterization of the method.

159

6.3.1 Phase Kickback and Phase Estimation

Needless to say, unitary operations play a vital role in quantum mechanics, particularly quan-

tum computing. A fundamental computational task one might be interested in is estimating the

eigenvalues of a unitary 𝑈. Given a quantum state |𝜓⟩ on the Hilbert space of 𝑈, one way to learn

the eigenvalues of 𝑈 is through a projective measurement in the eigenbasis. This has the added

benefit of approximately preparing a corresponding eigenstate.

Phase estimation algorithms accomplish precisely this goal. For readers familiar with basic op-

tics, a satisfying analogy exists between phase estimation and Mach-Zehnder interferometers [67].

In fact, the analogy is so close that it is more accurate to say they share the same working principle:

measuring a phase shift via interference.

Phase estimation algorithms have been around for as long as quantum computing has garnered

significant attention. Shor’s famous algorithm for factoring [118], and the more general problem

of finding discrete logarithms, rests on phase estimation techniques, as does the HHL algorithm

for solving linear systems [64]. Standard QPE, based on the Quantum Fourier Transform, is

described in detail in Nielsen and Chuang’s well-known text [101]. Kitaev supplied perhaps

the first iterative QPE protocol [77], and several improvements to the method have been made

since [125, 67, 102]. Adaptive protocols allow for improved phase measurement schemes based

on prior measurements [136]. Often iterative QPE refers simply to a "iteratization" of the standard

QFT by pushing all controls past the measurements [59].

6.3.2 Basic Circuit

Figure 6.1 exhibits the fundamental iteration of the Rodeo Algorithm, which we term a "cycle."

The upper wire represents a single auxiliary qubit, on which single qubit gates such as the Hadamard

𝐻 and the parametrized phase gate

𝑆(𝛼) := (cid:169)
(cid:173)
(cid:173)
(cid:171)
act. The lower register is the main register of interest, where 𝑈 (𝑡) = 𝑒−𝑖𝑂𝑡 is the time evolution

0 𝑒𝑖𝛼

(6.26)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

1

0

operator for some observable 𝑂 of interest. The parameter 𝐸 is called the target energy, and set

160

|0⟩

𝐻

𝑆(𝐸𝑡)

𝐻

|𝜓0⟩

𝑈 (𝑡)

Figure 6.1 Elementary iteration ("cycle") of the Rodeo Algorithm. Times 𝑡 are randomly sampled
from a normal distribution of center 0 and width Γ−1. Performing 𝑀 cycles acts as a band-pass
filter, only allowing eigenvalues within a range centered around 𝐸 with width Γ. Eigenvalues
outside this interval are exponentially suppressed in the number of cycles 𝑀. "Success" is
conditioned on all 𝑀 measurement outcomes being 0.

by the user. Meanwhile, 𝑡 is a random variable sampled from some distribution 𝜌 centered about

zero. We take 𝑡 to be normally distributed with variance 1/Γ2.

𝜌(𝑡) =

Γ
√
2𝜋

𝑒−(Γ𝑡)2/2

(6.27)

Other reasonable choices exist, but the normal distribution is simple to analyze and performs well

enough. Some choices, such as the uniform distribution over [−Γ−1, Γ−1], are less favorable as they

have poorer filtering properties resulting from the distribution having long tails in Fourier (energy)

space.

Roughly speaking, the parameters 𝐸, Γ define an interval [𝐸 − Γ, 𝐸 + Γ] for which the Rodeo

measurement protocol asks the question: Is there an eigenvalue of 𝑂 located in [𝐸 − Γ, 𝐸 + Γ]? Like

all quantum measurements, this will depend on the state |𝜓0⟩ prepared, and a successful detection

will occur with frequency given by the Born rule. As Γ shrinks, longer time evolutions 𝑈 (𝑡) will

occur, and this is expensive. Taking the cost to increase linearly in 𝑡, which saturates lower bounds

of the no-fast-forwarding theorem [14], the cost scales inversely with the accuracy. This is in accord

with the Heisenberg limit for quantum parameter estimation.

The role of the phase gate 𝑆(𝐸𝑡) is to shift the spectrum of 𝑂 by −𝐸. This is because the phase

acts equivalently to a controlled phase multiplication on the system.

𝑆(𝛼)

=

𝑒𝑖𝛼

Applying this identity to the circuit of Figure 6.1 and combining the control gates,

161

𝐻

𝑆(𝐸𝑡)

𝐻

𝐻

𝐻

𝑈 (𝑡)

=

𝑈𝐸 (𝑡)

where

𝑈𝐸 (𝑡) = 𝑒𝑖𝐸𝑡𝑈 (𝑡) = 𝑒−𝑖(𝑂−𝐸 𝐼)𝑡 .

(6.28)

For eigenvalues 𝜆 of 𝑂 within 𝑂 (Γ) of 𝐸, 𝑒−𝑖(𝜆−𝐸)𝑡 is relatively unaffected by variations in 𝑡 of

order 𝑂 (Γ−1). Meanwhile, other eigenvalues are shifted dramatically as 𝑡 varies randomly. This

we analogize as the "bucking" in the Rodeo Algorithm, where far-away eigenvalues are likely to be

kicked off. This will be elucidated more concretely in subsequent analysis.

As far as the author is aware, our algorithm is the first of the phase estimation family to employ

random parameters and shift the Hamiltonian in this fashion. Even if our initial state |𝜓0⟩ has small

overlap with the eigenstates we are interested in, successive iterations of this circuit, with 𝐸 and Γ

trained on the energy range of interest, allow us to amplify these states and determine whether our

operator 𝑂 has some eigenvalue in the range set by these parameters. We do require |𝜓0⟩ to have

some overlap with these eigenstates. However, the threshold for detection can be made increasingly

small with repeated cycles.

6.3.3 A Single Buck of the Bull

Let |𝜓0⟩ be the initial state of the main register, as in Figure 6.1. We decompose |𝜓0⟩ into its

spectral components in the following way.

𝛼∈𝜎(𝑂)
Here, 𝜎(𝑂) is the spectrum of 𝑂, i.e., the set of (real) eigenvalues, 𝑐𝛼 ∈ C is the component of

|𝜓0⟩ =

∑︁

𝑐𝛼 |𝛼⟩

(6.29)

|𝜓0⟩ along the eigenspace of 𝛼, and |𝛼⟩ is the projection onto this subspace.

|𝛼⟩ :=

𝑃𝛼 |𝜓⟩
√︁⟨𝜓| 𝑃𝛼 |𝜓⟩

,

𝑐𝛼 := ⟨𝛼| 𝑃𝛼 |𝜓⟩ = √︁⟨𝜓| 𝑃𝛼 |𝜓⟩

(6.30)

We will suppress the notation 𝜎(𝑂) in the sum (6.29) from now on. The output of the rodeo cycle,

before measurement, is given by the well-known result for the Hadamard test.

|0⟩ |𝜓0⟩ → |0⟩

(cid:16) 𝐼 + 𝑈𝐸 (𝑡)
2

(cid:17)

|𝜓0⟩

+ |1⟩

(cid:16) 𝐼 − 𝑈𝐸 (𝑡)
2

(cid:17)

|𝜓0⟩

(6.31)

162

Expressing this in terms of the eigenbasis |𝛼⟩ gives

|0⟩ |𝜓0⟩ → |0⟩ ⊗

+ |1⟩ ⊗

∑︁

𝛼
∑︁

𝛼

𝑐𝛼𝑒−𝑖Δ𝛼𝑡/2 cos(

𝑐𝛼𝑖𝑒−𝑖Δ𝛼𝑡/2 sin(

Δ𝛼𝑡
2
Δ𝛼𝑡
2

) |𝛼⟩

) |𝛼⟩

(6.32)

where we have defined Δ𝛼 = 𝛼 − 𝐸. If the measurement succeeds (0), we continue on with the

iteration of the circuit and preserve the state of the main register, while for failure we either halt

or discard the result at the end of the computation. Assuming success, the new state vector |𝜓′⟩ is

given by

|𝜓′⟩ =

∑︁

𝛼

𝑐′
𝛼 |𝛼⟩

𝛼 = 𝑐𝛼𝑒−𝑖Δ𝛼𝑡/2 cos
𝑐′

(cid:19)

(cid:18) Δ𝛼𝑡
2

up to normalization. The probability 𝑝′

𝛼 = |𝑐′

𝛼|2 is slightly more illuminating.

𝑝′
𝛼 = 𝑝𝛼 cos2

(cid:19)

(cid:18) Δ𝛼𝑡
2

(6.33)

(6.34)

Observe that a relative enhancement of the probability amplitudes 𝑝𝛼 occurs for cos2(Δ𝛼𝑡/2) ≈ 1.

This certain occurs for Δ𝛼𝑡 ≈ 0, i.e., for 𝛼 near the target energy, but also for Δ𝛼𝑡 = 2𝜋𝑘 with

𝑘 ∈ Z. These peak locations for 𝑘 ≠ 0 fluctuate with 𝑡. With many cycles, it is unlikely that any

eigenvalue far from 𝐸 will stay on a maximum across multiple trials, as we will see shortly.

The probability of success 𝑃0 and failure 𝑃1 is computed from the squared norm of each term

in equation (6.32).

𝑃0 =

𝑃1 =

∑︁

𝛼
∑︁

𝛼

𝑝𝛼 cos2

𝑝𝛼 sin2

(cid:19)

(cid:19)

(cid:18) Δ𝛼𝑡
2
(cid:18) Δ𝛼𝑡
2

= 1 − 𝑃0

(6.35)

6.3.4 Multiple Bucks: the Full Rodeo

The extension of the previous analysis from a single run to 𝑀 runs through the basic circuit

of Figure 6.1 is relatively straightforward. Let (𝑡𝑖) 𝑀

𝑖=1 be the time samples for each cycle. Then,

163

with repeated application of equation (6.34), the probability amplitude 𝑝 (𝑀)

𝛼

of 𝛼 after 𝑀 runs,

conditioned on the circuit succeeding, is given by

𝑝 (𝑀)
𝛼 = 𝑝𝛼

𝑀
(cid:214)

𝑖=1

cos2

(cid:19)

.

(cid:18) Δ𝛼𝑡𝑖
2

The success probability 𝑃(𝑀)

0 may be factorized as

𝑃(𝑀)
0

=

𝑀
(cid:214)

𝑘=1

𝑍𝑘

(6.36)

(6.37)

where 𝑍𝑘 is the probability of measuring zeros on the 𝑘th measurement conditioned on measuring
zeros in every prior measurement. Let |Ψ𝑘−1⟩ = |0⟩ ⊗ (cid:205)𝛼 𝑐(𝑘−1)
after 𝑘 − 1 successful measurements and directly before the 𝑘th measurement. From Born’s rule,

|𝛼⟩ be the state of the entire register

𝛼

𝑍𝑘 =

∥ ⟨0|Ψ𝑘−1⟩ ∥2
⟨Ψ𝑘−1|Ψ𝑘−1⟩

,

where ⟨0| ≡ ⟨0| ⊗ 𝐼. Adapting equation (6.32) to the present situation,

𝑍𝑘 =

(cid:205)𝛼 𝑝 (𝑘−1)
𝛼

cos2 (Δ𝛼𝑡𝑘 /2)

(cid:205)𝛽 𝑝 (𝑘−1)
𝛽

=

(cid:205)𝛼 𝑝 (𝑘)
𝛼
(cid:205)𝛽 𝑝 (𝑘−1)
𝛽

.

(6.38)

(6.39)

Returning to expression (6.37), we see that each 𝑍𝑘 in the product telescopes, giving a simple

formula.

𝑃(𝑀)
0

∑︁ 𝑝 (𝑀)

𝛼 =

=

∑︁

𝑝𝛼

𝛼

𝑀
(cid:214)

𝑘=1

cos2

(cid:19)

(cid:18) Δ𝛼𝑡𝑘
2

(6.40)

That is, the success probability is simply the sum of the unnormalized probability amplitudes. In

hindsight, we could have anticipated that, for each 𝛼, the measurement probabilities over each

iteration behave independently, as exhibited in equation (6.40). Performing the analysis for an

eigenvector input state, one finds that the state is unaffected by the measurement outcomes, thus

the measurements are independent. The full result follows by linearity.

6.3.5 Statistics

Now that we’ve determined the behavior for a particular choice of times (𝑡𝑖) 𝑀

𝑖=1, we must recall

that these times were randomly chosen according to the normal distribution (6.27). We hope,

164

though have yet to fully justify, that the measurement statistics correlate strongly, and predictably,

with the presence of an eigenvalue within 𝐸 ± Γ that significantly overlaps the initial state. To do

this, it makes sense to compute some basic statistics about the measurement results, particularly the
expected behavior over the distribution of 𝑡𝑖. Thus, we compute the expectation values of 𝑝 (𝑀)
and P (𝑀)
0

(𝑡). Using equation (6.36),

(𝑡)

𝛼

⟨𝑝 (𝑀)
𝛼

⟩ =

∫

R𝑀

𝜌(𝑡) 𝑝 (𝑀)

𝛼 𝑑𝑡 𝑀

=

=

Γ
(2𝜋) 𝑀/2

𝑝𝛼Γ
(2𝜋) 𝑀/2

∫

R𝑀

(cid:18)∫

R

𝑒−(Γ𝑡)2/2 𝑝𝛼

𝑀
(cid:214)

cos2(

𝑒−(Γ𝑡)2/2 cos2

𝑗=1
(cid:18) Δ𝛼𝑡
2

(cid:19) 𝑀

(cid:19)

𝑑𝑡

.

Δ𝛼𝑡 𝑗
2

)𝑑𝑡 𝑀

(6.41)

In the last step, we used the fact that the 𝑀-dimensional integral factorizes. This expression is easy

to evaluate.

⟨𝑝 (𝑀)
𝛼

⟩ = 𝑝𝛼

(cid:33) 𝑀

(cid:32)

1 + 𝑒−Δ𝛼2/2Γ2
2

(6.42)

We see that, for 𝛼 − 𝐸 on the order of Γ or greater, the probability decays in the number of cycles

√

as 2−𝑀, whereas for
success probability ⟨𝑃(𝑀)

0

𝑀Δ𝛼/Γ ≪ 1 the amplitudes are approximately preserved. The expected

⟩ can be easily obtained from equations (6.40) and (6.42), using the

linearity of the expectation value.

⟨𝑃(𝑀)
0

⟩ =

∑︁

𝑝𝛼

𝛼

(cid:33) 𝑀

(cid:32)

1 + 𝑒−Δ𝛼2/2Γ2
2

(6.43)

Observe how this serves as an indicator function for the existence of eigenvalues. If all 𝛼 are outside

of 𝑂 (Γ) from 𝐸, the amplitudes decay exponentially in 𝑀. For any 𝛼 within this range, however,

there will be a success probability which goes roughly as the initial overlap with those states. As

expected from measurement-based procedures, we cannot overcome the inherent 𝑂 (1/overlap2)

cost scaling. A quadratic improvement to a Heisenberg limit may be feasible with amplitude

amplification, but only with additional quantum overhead and operations.

Our indicator ⟨𝑃(𝑀)

0

⟩ might not be of practical use if the behavior of a typical run deviates

165

wildly from the average. We thus investigate the variance. From equation (6.40),

Var(𝑃(𝑀)

0

(cid:68) (cid:0)𝑃(𝑀)

0

) =
= (cid:10) ∑︁

(cid:1) 2(cid:69)

−

(cid:69)2

(cid:68)

𝑃(𝑀)
0

𝛼 𝑝 (𝑀)
𝑝 (𝑀)
𝛽

(cid:11) − (cid:0) ∑︁

⟨𝑝 (𝑀)
𝛼

⟩(cid:1) 2

𝛼𝛽

Cov

(cid:16)

𝑝 (𝑀)
𝛼

, 𝑝 (𝑀)
𝛽

=

∑︁

𝛼𝛽

𝛼

,

(cid:17)

(6.44)

where

Cov (𝑋, 𝑌 ) := ⟨𝑋𝑌 ⟩ − ⟨𝑋⟩⟨𝑌 ⟩

(6.45)

is the covariance. To clean up the math below, define the dimensionless parameters 𝑎 := Δ𝛼/Γ and

𝑏 := Δ𝛽/Γ. Then,

⟨𝑝 (𝑀)

𝛼 𝑝 (𝑀)
𝛽

⟩ = 𝑝𝛼 𝑝 𝛽

⟨𝑝 (𝑀)
𝛼

⟩⟨𝑝 (𝑀)
𝛽

⟩ = 𝑝𝛼 𝑝 𝛽

(cid:32)

(cid:32)

2 + 𝑒−(𝑎+𝑏)2/2 + 𝑒−(𝑎−𝑏)2/2 + 2𝑒−𝑎2/2 + 2𝑒−𝑏2/2
8

(cid:33) 𝑀

1 + 𝑒−(𝑎2+𝑏2)/2 + 𝑒−𝑎2/2 + 𝑒−𝑏2/2
4

(cid:33) 𝑀

so that

where

Cov

(cid:16)

𝑝 (𝑀)
𝛼

, 𝑝 (𝑀)
𝛽

(cid:17)

= 𝑝𝛼𝐶 (𝑀)

𝛼𝛽 𝑝 𝛽

(cid:32)

(cid:32)

𝐶 (𝑀)

𝛼𝛽 =

−

1 + 𝑒−𝑎2/2 + 𝑒−𝑏2/2 + 𝑒−(𝑎2+𝑏2)/2 cosh(𝑎𝑏)
4

(cid:33) 𝑀

1 + 𝑒−𝑎2/2 + 𝑒−𝑏2/2 + 𝑒−(𝑎2+𝑏2)/2
4

(cid:33) 𝑀

(6.46)

(6.47)

(6.48)

is positive definite. Hence, the variance Var(P (𝑀)

0

) is a contraction of the matrix 𝐶 (𝑀) with the

vector of initial probability amplitudes 𝑝 for each eigenvalue.

It is fruitful to consider 𝐶 (𝑀)

𝛼𝛽 as a function of two real parameters 𝑎, 𝑏 ∈ R for each 𝑀 ∈ Z+.
First, we observe that 𝐶 is an even function in both 𝑎 and 𝑏, so that only the positive quadrant need

be considered. We also observe that 𝐶 is symmetric under 𝑎 ↔ 𝑏. It is more or less clear that

𝐶 should approach 0 for 𝑎, 𝑏 large and for 𝑎, 𝑏 near zero, but showing this analytically is rather
awkward (though straightforward in principle). Figure 6.2 provides plots of 𝐶 (𝑀)

𝛼𝛽 for various values

166

Figure 6.2 (Top) Density plot of covariance function 𝐶 (𝑀)
𝑏 = Δ𝛽/Γ for 𝑀 = 4, 10, and 20 going left to right. We observe the function diminishing with 𝑀,
with maximum along the 𝑎 = 𝑏 line which moves slightly inward with 𝑀. (Bottom) Location and
values of maximum 𝐶 (𝑀)
for 𝑀 from 4 to 200. Line of best fit indicates inverse square root power
𝛼𝛽
law for location, whereas value 𝑐max appears to follow inverse power law with 𝑀.

𝛼𝛽 with respect to 𝑎 = Δ𝛼/Γ and

of 𝑀. We observe the correlations are peaked for 𝑎 = 𝑏 = 𝑎peak, where 𝑎peak appears to follow an

inverse square root power law. Even for 𝑀 = 4, the correlations are never larger than 0.06 and only

decrease with 𝑀.

The peak location corresponds to eigenvalues being on the order of Θ(Γ) away from the target

energy. Larger values of 𝐶 (𝑀)
𝛼𝛽

contribute to a larger variance via (6.44). We interpret this as

follows: the Rodeo Algorithm struggles to properly classify eigenvalues close to, but not entirely

within, the rough interval [𝐸 − Γ, 𝐸 + Γ]. When eigenvalues are clearly within Γ, the success

probability is 1 for those eigenstates, while for far eigenvalues the success probability is a coin toss.

This "resonant peak", despite being a nuisance, can be handled by varying the target energy, and

in any case the resonance decreases in amplitude and width with 𝑀. We will make use of some
properties of 𝐶 (𝑀)
𝛼𝛽

in our subsequent analysis of the Rodeo Algorithm’s performance.

167

6.3.6 Performance on Eigenvalue Detection

Having characterized the statistical properties of our randomized circuit, we now turn to the

question of performance. To make progress analytically, we make some simplifying assumptions.

Suppose we choose some target energy 𝐸 and width Γ, and wish to determine whether there is

a nearby eigenvalue of 𝛼∗, by which we mean an eigenvalue such that |𝛼∗ − 𝐸 |/Γ < 𝑑 for 𝑑 on

the order of 1. We are promised that, if such an eigenstate exists, the input state |𝜓⟩ has overlap

𝑝𝛼∗ = |𝑐𝛼∗ |2 ≥ 𝛿, where 𝛿 > 0 is some given threshold. Any other populated eigenstates are
assumed to have eigenvalues at least 𝑔Γ away from 𝐸 for some 𝑔 > 𝑑.

We analyze the question of eigenvalue existence in the language of hypothesis testing, with null

hypothesis 𝐻null of no eigenvalue present. The alternative hypothesis 𝐻alt is the presence of 𝛼∗

within Γ𝑑 of 𝐸 that has overlap at least 𝛿 with the initial state. Practically speaking, without the

promises given above, our algorithm will simply fail to detect eigenstates whose overlap is too low

(without the 𝛿 promise), or fail to resolve multiple eigenvalues within the detection range (without

the 𝑔 promise).

Under 𝐻0, the expected success probability 𝑃null of the 𝑀-cycle Rodeo Algorithm is upper

bounded as

𝑃null ≤

(cid:32)

1 + 𝑒−𝑔2/2
2

(cid:33) 𝑀

.

(6.49)

On the other hand, for alternative hypothesis 𝐻1, the expected success probability 𝑃alt is lower

bounded as

𝑃alt ≥ 𝛿

> 𝛿

(cid:32)

(cid:32)

1 + 𝑒−𝑑2/2
2

1 + 𝑒−𝑑2/2
2

(cid:33) 𝑀

+ (1 − 𝛿)

1
2𝑀

(cid:33) 𝑀

.

(6.50)

To distinguish the two cases, we estimate the success probability through a normalized count 𝑁

samples of the 𝑀 cycle Rodeo Algorithm. A detection corresponds to determining 𝑃alt > 𝑃null

with confidence determined by uncertainty in the method. We denote the estimated probability by

¯𝑃. Ignoring other reasonable sources of error, such as imperfect gates, decoherence, and imperfect

implementation of 𝑈 (𝑡), the error in the estimate is 𝑂 (𝜎sample), where 𝜎sample is the standard

168

deviation in the estimate of expected success probability. To analyze this uncertainty, we use a

heuristic error propagation approach. Assuming an exact, nonrandom success probability 𝑃0, our

uncertainty comes from the binomial variance

𝜎binom =

√︂ 𝑃0(1 − 𝑃0)
𝑁

<

√︂ 𝑃0
𝑁

.

(6.51)

To characterize deviations from the binomial distribution due to random fluctuations 𝜎fluc of 𝑃0

caused by the times 𝑡 𝑗 , we use an error propagation formula.

𝜎fluc ≈

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝜕
𝜕𝑃0

√︂ 𝑃0(1 − 𝑃0)
𝑁

𝜎𝑃0

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

< 1
2

√

1
𝑃0𝑁

𝜎𝑃0

(6.52)

Let’s now consider the two cases separately, in the case of 𝐻null, there is no eigenvalue within 𝑔Γ

and the upper bound

𝜎fluc < 1
√
𝑁
is valid. In this case, 𝜎𝑃0 will shrink exponentially, and for some 𝑔 ∈ 𝑂 (1) this will be enough to
bound 𝜎fluc by 𝑂 (√︁𝑃0/𝑁). Consider now the case 𝐻alt. From now on, we make the assumption

2𝑀/2𝜎𝑃0

(6.53)

𝑑2𝑀 ≪ 1

(6.54)

so that 𝑃alt > 𝛿 + 𝑂 (𝛿𝑑2𝑀). In this case, we require 𝜎𝑃0
of 𝐶 (𝑀)
guarantee an 𝑂 (√︁𝑃0/𝑁) scaling as desired.

𝛼𝛽 , particularly the 𝛼 = 𝛽 = 𝑑Γ + 𝐸 term, reveals that 𝐶 (𝑀)

< √︁𝛿 + 𝑂 (𝛿𝑑2𝑀). A careful analysis

𝛼𝛽 ∈ 𝑂 (𝑑4𝑀). This is sufficient to

Overall, we’ve found, through semi-heuristic derivation, that the error in our Rodeo estimation

protocol goes as 𝑂 (1/

√

𝑁). To ensure the two hypotheses can be distinguished by the Rodeo

Algorithm, we thus require

(cid:32)

𝛿

1 + 𝑒−𝑑2/2
2

(cid:33) 𝑀

(cid:32)

−

1 + 𝑒−𝑔2/2
2

(cid:33) 𝑀

∈ Ω(√︁𝛿/𝑁).

Taking as the worst case scenario 𝑔 = 1 in the above, we have the requirement

(cid:33) 𝑀

(cid:32)

1 + 𝑒−𝑑2/2
2

∈ Ω

(cid:18) 1
√
𝛿𝑁

+

1
𝛿

𝑒−Θ(𝑀)

(cid:19)

.

169

(6.55)

(6.56)

We utilize our requirement that 𝑑2𝑀 ≪ 1, so that the right-hand side is Ω(1). This is implicitly

an upper bound on 𝑀, meaning that not all 𝑑 and 𝛿 will allow for solutions using our approach.

However, we will see presently that the size of 𝑀 need not be too large. Looking at the 2nd term

on the right-hand size of (6.56), we find that

𝑀 ∈ Ω(log 1/𝛿)

(6.57)

suffices to ensure 𝑂 (1) error. Finally, we choose 𝑁 ∈ Ω(1/𝛿).

We see that there are choices of parameters for which the algorithm can succeed in the setting

we’ve constructed. The only real requirement is that

log 1/𝛿 ≪ 1/𝑑2

(6.58)

which is not a great restriction in practice, provided the overlap is not exceedingly small. This

restriction comes about, more or less, from the fact that, as 𝑀 increases, the effective width of the

search window shrinks as 𝑂 (1/

√

𝑀).

A rough cost estimate 𝐶ost can be assigned as

𝐶ost :=

𝑁 𝑀

Γ

∈ 𝑂

(cid:18) (1/𝛿) log 1/𝛿
Γ

(cid:19)

(6.59)

which attributes a cost of Γ−1 to the time evolution, assuming it dominates the full rodeo cycle.

The cost 𝑐ost = 𝐶ost/𝑁 per cycle is also the maximum circuit depth, and has the same Γ scaling but

favorable 𝛿 dependence. Interpreting Γ as our accuracy parameter, we see that the Rodeo Algorithm

achieves Heisenberg scaling.

In practice, given a range [𝐸min, 𝐸max] of possible eigenvalues for the operator 𝑂 of interest, Γ

can be varied using a procedure akin to binary search. This has little effect on the total cost, which

will still be dominated by the smallest Γ used assuming it is reduced exponentially each refinement.

6.3.7 Noise

Noise and decoherence are significant sources of error for near-term quantum computers without

error correction. Significant advances in error mitigation techniques have increased what can be

achieved from limited, noisy hardware [76], though there may be fundamental limitations to these

170

approaches for large systems [108]. In search of applications for near-term quantum devices, there

is a desire for methods which are inherently robust to noise.

We investigated the noise robustness of the Rodeo Algorithm under the simplest possible model

of decoherence: depolarizing noise. At each gate, we assume there is some probability 𝑝depol of the

current state being replaced with the maximally mixed state, thus ruining the computation. Compu-

tations were performed using Qiskit Runtime with the "ibmq_qasm_simulator." We considered the

simple 3-qubit Hamiltonian 𝑍1 + 𝑍2 + 𝑍3, whose eigenvalues are easily seen to be the odd integers

from −3 to 3. We took as initial state (𝐻 |0⟩)⊗3 where 𝐻 is the Hadamard. We varied 𝑝depol from 0

to 0.05 and investigated the effect on performance. We take Γ = 1 and 𝑀 = 6, and perform a scan

from [−5, 5] in increments of 0.1. Figure 6.3 shows the results of the numerical simulation. We

observe that, for increased noise, the peaks decrease in height but remain in the same location. By

𝑝depol = 0.05, the peaks become hard to distinguish from one another, but not hard to distinguish

from background. Importantly, the location of the peaks is relatively unaffected, which is the most

important part. We expect that, for many reasonable models of noise, not just symmetric depolar-

ization, the peaks will not shift greatly as a result. Imperfect gates will change the effective operator

˜𝑂, however, thus leading to changes in eigenvalues. Understanding how decoherence affects the

effective Hamiltonian being evolved would greatly advance our understanding of the impact, not

only on the Rodeo Algorithm, but other phase-estimation protocols.

6.3.8 Discussion

Here we presented the Rodeo Algorithm, a new addition to the collection of phase estimation

protocols which is simple and allows for targeted search of eigenvalues and preparation of eigen-

states. To complement previous work on the method [29, 106, 11], here we focused on theoretical

aspects not previously covered. We provide a quasi-rigorous cost analysis of the method under a

binary hypothesis model, and conclude with some numerics showing behavior of the method under

symmetric depolarizing noise.

Following the initial publication of the method, subsequent research removed the need for

randomness in the algorithm while providing a more rigorous characterization of performance [96].

171

Figure 6.3 Simulation of Rodeo Algorithm for three non-interacting qubits for various amounts of
depolarizing noise. Eigenvalues are located at −3, −1, 1, and 3, and the initial state has overlap
1/8, 3/8, 3/8, 1/8 with each eigenspace. We see that the algorithm reproduces the correct
qualitative behavior in spite of noise reductions in signal. The algorithm benefits from the
symmetry of depolarization, whereas other noise models may affect the phase rotation being
applied.

These authors were interested particularly in state preparation, and their findings of performance

are consistent with our randomized approach.

For current noisy devices, the Rodeo Algorithm can serve as a simple and practical alternative

for standard QPE for eigenvalue estimation and state preparation. Thus, it may eventually serve as a

useful subroutine for algorithms aimed at achieving quantum advantage, that is, practical advantage

of quantum computers over classical computers for interesting tasks. For example, recently an end-

to-end simulation algorithm for nuclear effective field theories was proposed, which used standard

Quantum Phase Estimation (QPE) in analysis for estimating total resource costs [134]. Replacing

this final measurement step with the Rodeo Algorithm in actual applications should increase the

feasibility of the full algorithm for near-term hardware.

172

CHAPTER 7

CONCLUSION AND OUTLOOK

Quantum computing today shares similarities with thermodynamics of the 19th century. From its

birth, thermodynamics was intimately linked to technology. Practitioners sought to create more

efficient engines, refrigerators, and other cyclical systems to perform work using the resource of

heat gradients. These pragmatic goals led to foundational scientific progress, such as discovering

laws about optimal efficiencies and allowed physical transformations. Gradually, the powerful and

subtle notion of entropy developed, so fruitful as to remain at the forefront of modern physics and

information theory.1

Quantum computers, unlike engines and refrigerators, have yet to yield any practical benefit.

Interest in the field is driven by the expectation that they eventually will, and that effective, error-

corrected quantum computers can be made that will simulate quantum dynamics, solve optimization

problems, and perform hitherto unrecognized yet valuable computational tasks. In the background,

deeper questions are very much present. How different are the quantum and classical worlds

from a computational viewpoint? What makes quantum probabilities special compared to standard

probability? Broadly, what is quantum mechanics even about? Because the building blocks of

quantum information, qubits, are so simple, they provide an excellent playground to tackle such

questions with clarity. It is imaginable that large-scale manipulation of entanglement will aid our

"gut" understanding of the theory and elucidate connections between the microscopic world and

our macroscopic reality.

For what purposes will we manipulate this enormous entanglement? Besides Shor’s ground-

breaking factoring algorithm, Grover’s unstructured search algorithm, and a handful of others, not

many compelling use cases are known. The truth is, if a scalable, fault-tolerant quantum computer

existed today, we wouldn’t know many uses for it. The idea of quantum-based quantum simulation

has been around about as long as quantum computing itself, and remains one of the few concrete and

useful tasks we know of. Thankfully, its importance to scientific computing is enough to command

1While the "entropy" from physics and information theory are technically distinct, Landauer’s principle [81] and

other ideas suggest a deep connection that, to my knowledge, is still not fully understood.

173

attention, and moreover, Hamiltonian simulation finds application in at least a handful general

tasks such as solving linear systems of equations. Even if not part of an algorithm, Hamiltonian

simulation concepts can assist in the design of quantum algorithms, especially given the absence

of many other conceptual guideposts.

This dissertation offers additional tools for Hamiltonian simulation and, importantly, rigorous

characterizations of each of their performance. We demonstrate increased effectiveness of prod-

uct formulas for accurately estimating observable dynamics using polynomial interpolation. We

provide several new approaches to time dependent Hamiltonian simulation, as well as a compu-

tational reduction of the time dependent to time independent dynamics. Finally, we survey novel

tools for near-term quantum simulation, including trapped-ion simulation of anomalous symmetry

breaking, state preparation via Projected Cooling, and resource-effective eigenvalue estimation and

eigenvector preparation using the Rodeo Algorithm.

Despite enormous progress, fresh ideas are needed in quantum algorithms research. We offer

tools which may eventually be implemented on the quantum computers of the future, and hope that

that future is not so far away.

174

BIBLIOGRAPHY

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

Scott Aaronson. Quantum computing and hidden variables. Physical Review A, 71(3):
032325, 2005.

Scott Aaronson. How big are quantum states?, page 200–216. Cambridge University Press,
2013.

Andrew Adamatzky. A brief history of liquid computers. Philosophical Transactions of the
Royal Society B, 374(1774):20180372, 2019.

Dorit Aharonov, Wim Van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev.
Adiabatic quantum computation is equivalent to standard quantum computation. SIAM
Review, 50(4):755–787, 2008.

Thomas D Ahle. Sharp and simple bounds for the raw moments of the binomial and poisson
distributions. Statistics & Probability Letters, 182:109306, 2022.

Rizwanul Alam, George Siopsis, Rebekah Herrman, James Ostrowski, Phillip C Lotshaw,
and Travis S Humble. Solving maxcut with quantum imaginary time evolution. Quantum
Information Processing, 22(7):281, 2023.

Alain Aspect, Jean Dalibard, and Gérard Roger. Experimental test of bell’s inequalities using
time-varying analyzers. Physical Review Letters, 49(25):1804, 1982.

Brian M Austin, Dmitry Yu Zubarev, and William A Lester Jr. Quantum monte carlo and
related approaches. Chemical Reviews, 112(1):263–288, 2012.

[9] Mohsen Bagherimehrab, Yuval R Sanders, Dominic W Berry, Gavin K Brennen, and Barry C
Sanders. Nearly optimal quantum algorithm for generating the ground state of a free quantum
field theory. PRX Quantum, 3(2):020364, 2022.

[10] Christian W. Bauer, Plato Deliyannis, Marat Freytsis, and Benjamin Nachman. Practical
considerations for the preparation of multivariate gaussian states on quantum computers,
2021.

[11] Max Bee-Lindgren, Zhengrong Qian, Matthew DeCross, Natalie C. Brown, Christopher N.
Gilbreth, Jacob Watkins, Xilin Zhang, and Dean Lee. Controlled gate networks applied to
eigenvalue estimation, 2024.

[12]

John S Bell. On the einstein podolsky rosen paradox. Physics Physique Fizika, 1(3):195,
1964.

[13] Daniel Berend and Tamir Tassa. Improved bounds on bell numbers and on moments of sums
of random variables. Probability and Mathematical Statistics, 30(2):185–205, 2010.

175

[14] Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. Efficient quantum
algorithms for simulating sparse hamiltonians. Communications in Mathematical Physics,
270:359–371, 2007.

[15] Dominic W Berry, Andrew M Childs, and Robin Kothari. Hamiltonian simulation with
nearly optimal dependence on all parameters. In 2015 IEEE 56th annual symposium on
foundations of computer science, pages 792–809. IEEE, 2015.

[16] Dominic W Berry, Andrew M Childs, Yuan Su, Xin Wang, and Nathan Wiebe. Time-

dependent hamiltonian simulation with l1-norm scaling. Quantum, 4:254, 2020.

[17] Alexandre Blais, Arne L Grimsmo, Steven M Girvin, and Andreas Wallraff. Circuit quantum

electrodynamics. Reviews of Modern Physics, 93(2):025005, 2021.

[18] S Blanes, F Casas, and J Ros. Extrapolation of symplectic integrators. Celestial Mechanics

and Dynamical Astronomy, 75:149–161, 1999.

[19] Colin D Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M Sage. Trapped-ion
quantum computing: Progress and challenges. Applied Physics Reviews, 6(2), 2019.

[20] Horacio E Camblong, Luis N Epele, Huner Fanchiotti, and Carlos A Garcia Canal. Quantum

anomaly in molecular physics. Physical Review Letters, 87(22):220402, 2001.

[21] Giulio Casati and Luca Molinari.

“quantum chaos” with time-periodic hamiltonians.

Progress of Theoretical Physics Supplement, 98:287–322, 1989.

[22] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo,
Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Varia-
tional quantum algorithms. Nature Reviews Physics, 3(9):625–644, 2021.

[23] Chi-Fang Chen and Fernando G. S. L. Brandão. Average-case speedup for product for-
doi: 10.1007/

mulas. Communications in Mathematical Physics, 405(2):32, 2024.
s00220-023-04912-5.

[24] Andrew M Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of
unitary operations. Quantum Information & Computation, 12(11-12):901–924, 2012.

[25] Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su. Toward the
first quantum simulation with quantum speedup. Proceedings of the National Academy of
Sciences, 115(38):9456–9461, 2018.

[26] Andrew M Childs, Aaron Ostrander, and Yuan Su. Faster quantum simulation by random-

ization. Quantum, 3:182, 2019.

[27] Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu. Theory of

176

trotter error with commutator scaling. Physical Review X, 11(1):011020, 2021.

[28] Siu A Chin. Multi-product splitting and runge-kutta-nyström integrators. Celestial Mechan-

ics and Dynamical Astronomy, 106(4):391–406, 2010.

[29] Kenneth Choi, Dean Lee, Joey Bonitati, Zhengrong Qian, and Jacob Watkins. Rodeo
algorithm for quantum computing. Physical Review Letters, 127(4):040505, 2021.

[30]

Juan I Cirac and Peter Zoller. Quantum computations with cold trapped ions. Physical
Review Letters, 74(20):4091, 1995.

[31] Louis Comtet. Advanced Combinatorics: The Art of Finite and Infinite Expansions. Springer

Science & Business Media, 2012.

[32] B. Jack Copeland. The Church-Turing Thesis. In Edward N. Zalta, editor, The Stanford
Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, summer 2020
edition, 2020.

[33] Christopher J Cramer. Essentials of Computational Chemistry: Theories and Models. John

Wiley & Sons, 2013.

[34] Peter Cromwell, Elisabetta Beltrami, and Marta Rampichini. The borromean rings. Mathe-

matical Intelligencer, 20(1):53–62, 1998.

[35] Toby S Cubitt, David Perez-Garcia, and Michael M Wolf. Undecidability of the spectral gap.

Nature, 528(7581):207–211, 2015.

[36] André Pierro de Camargo. On the numerical stability of newton’s formula for lagrange
interpolation. Journal of Computational and Applied Mathematics, 365:112369, 2020.
ISSN 0377-0427. doi: https://doi.org/10.1016/j.cam.2019.112369.

[37] David Deutsch. Quantum theory, the church–turing principle and the universal quantum
computer. Proceedings of the Royal Society of London. A. Mathematical and Physical
Sciences, 400(1818):97–117, 1985.

[38] Bryce Seligman Dewitt and Neill Graham. The Many-Worlds Interpretation of Quantum

Mechanics, volume 63. Princeton University Press, 2015.

[39] Paul Adrien Maurice Dirac. Quantum mechanics of many-electron systems. Proceedings of
the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical
Character, 123(792):714–733, 1929.

[40] Yulong Dong, Lin Lin, and Yu Tong. Ground-state preparation and energy estimation on
early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary
matrices. PRX Quantum, 3(4):040305, 2022.

177

[41] Vitaly Efimov. Energy levels arising from resonant two-body forces in a three-body system.

Physics Letters B, 33(8):563–564, 1970.

[42] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum-mechanical description

of physical reality be considered complete? Physical Review, 47(10):777, 1935.

[43]

Jens Eisert, Marcus Cramer, and Martin B Plenio. Colloquium: Area laws for the entangle-
ment entropy. Reviews of Modern Physics, 82(1):277, 2010.

[44] Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin, and Xiao Yuan. Mitigating algorithmic
errors in a hamiltonian simulation. Physical Review A, 99:012334, 1 2019. doi: 10.1103/
PhysRevA.99.012334.

[45] Paul K Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert.

Randomizing multi-product formulas for hamiltonian simulation. Quantum, 6:806, 2022.

[46] Richard P Feynman. Simulating physics with computers. In Feynman and Computation,

pages 133–153. CRC Press, 2018.

[47] Fulvio Flamini, Nicolo Spagnolo, and Fabio Sciarrino. Photonic quantum information

processing: a review. Reports on Progress in Physics, 82(1):016001, 2018.

[48] Edward Fredkin and Tommaso Toffoli. Conservative logic. International Journal of Theo-

retical Physics, 21(3):219–253, 1982. doi: 10.1007/BF01857727.

[49] Stuart J Freedman and John F Clauser. Experimental test of local hidden-variable theories.

Physical Review Letters, 28(14):938, 1972.

[50] Nicolai Friis, Oliver Marty, Christine Maier, Cornelius Hempel, Milan Holzäpfel, Petar
Jurcevic, Martin B Plenio, Marcus Huber, Christian Roos, Rainer Blatt, et al. Observation
of entangled states of a fully controlled 20-qubit system. Physical Review X, 8(2):021012,
2018.

[51] W Gautschi. How (un)stable are vandermonde systems? asymptotic and computational
analysis. In Lecture Notes in Pure and Applied Mathematics, pages 193–210. Marcel Dekker,
Inc, 1990.

[52]

Iulia M Georgescu, Sahel Ashhab, and Franco Nori. Quantum simulation. Reviews of
Modern Physics, 86(1):153, 2014.

[53] Elizabeth Gibney. The quantum gold rush. Nature, 574(7776):22–24, 2019.

[54] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value
transformation and beyond: exponential improvements for quantum matrix arithmetics.
In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing,

178

page 193–204, New York, NY, USA, 2019. Association for Computing Machinery. ISBN
9781450367059.

[55] Sheldon Goldstein. Bohmian Mechanics. In Edward N. Zalta, editor, The Stanford Ency-
clopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2021 edition,
2021.

[56] Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner. Iterative quantum ampli-
tude estimation. npj Quantum Information, 7:52, 2021. doi: 10.1038/s41534-021-00379-1.

[57] Lov Grover and Terry Rudolph. Creating superpositions that correspond to efficiently

integrable probability distributions, 2002.

[58] Erik J Gustafson. Projective cooling for the transverse ising model. Physical Review D, 101

(7):071504, 2020.

[59] Dipanjali Halder, V Srinivasa Prasannaa, Valay Agarawal, and Rahul Maitra.

Iterative
quantum phase estimation with variationally prepared reference state. International Journal
of Quantum Chemistry, 123(3):e27021, 2023.

[60]

Jack K. Hale and Hüseyin Koçak. Scalar Nonautonomous Equations, pages 107–132.
ISBN 978-1-4612-4426-4. doi: 10.1007/
Springer New York, New York, NY, 1991.
978-1-4612-4426-4\_4.

[61] Brian Hall. Quantum Theory for Mathematicians, volume 267 of Graduate Texts in Mathe-

matics. Springer-Verlag New York, 2013.

[62] H-W Hammer and Brian G Swingle. On the limit cycle for the 1/r2 potential in momentum

space. Annals of Physics, 321(2):306–317, 2006.

[63] Dylan Harley, Ishaun Datta, Frederik Ravn Klausen, Andreas Bluhm, Daniel Stilck França,
Albert Werner, and Matthias Christandl. Going beyond gadgets: The importance of scala-
bility for analogue quantum simulators. arXiv:2306.13739, 2023.

[64] Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems

of equations. Physical Review Letters, 103(15):150502, 2009.

[65] Loïc Henriet, Lucas Beguin, Adrien Signoles, Thierry Lahaye, Antoine Browaeys, Georges-
Olivier Reymond, and Christophe Jurczak. Quantum computing with neutral atoms. Quan-
tum, 4:327, 2020.

[66] Markus Heyl, Philipp Hauke, and Peter Zoller. Quantum localization bounds trotter errors in
digital quantum simulation. Science Advances, 5(4):eaau8342, 2019. doi: 10.1126/sciadv.
aau8342.

179

[67] Brendon L Higgins, Dominic W Berry, Stephen D Bartlett, Howard M Wiseman, and Geoff J
Pryde. Entanglement-free heisenberg-limited phase estimation. Nature, 450(7168):393–396,
2007.

[68] Nicholas J Higham. The numerical stability of barycentric lagrange interpolation.

IMA

Journal of Numerical Analysis, 24(4):547–556, 2004.

[69] Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies

the ’magic’for quantum computation. Nature, 510(7505):351–355, 2014.

[70]

Jacky Huyghebaert and Hans De Raedt. Product formula methods for time-dependent
schrodinger problems. Journal of Physics A: Mathematical and General, 23(24):5777,
1990.

[71]

Jason Iaconis, Sonika Johri, and Elton Yechao Zhu. Quantum state preparation of normal
distributions using matrix product states. npj Quantum Information, 10(1):15, 2024.

[72] R Islam, Crystal Senko, Wes C Campbell, S Korenblit, J Smith, A Lee, EE Edwards, C-
CJ Wang, JK Freericks, and C Monroe. Emergence and frustration of magnetism with
variable-range interactions in a quantum simulator. Science, 340(6132):583–587, 2013.

[73] Dollard John Day and Friedman Charles N. Product Integration with Application to Differ-
ential Equations. Number v. 10. Section, Analysis in Encyclopedia of Mathematics and Its
Applications. Cambridge University Press, 1984. ISBN 9780521302302.

[74]

John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ron-
neberger, Kathryn Tunyasuvunakool, Russ Bates, Augustin Žídek, Anna Potapenko, et al.
Highly accurate protein structure prediction with alphafold. Nature, 596(7873):583–589,
2021.

[75] Mária Kieferová, Artur Scherer, and Dominic W Berry. Simulating the dynamics of time-
dependent hamiltonians with a truncated dyson series. Physical Review A, 99(4):042314,
2019.

[76] Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout Van Den Berg, Sami
Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, et al. Evidence for
the utility of quantum computing before fault tolerance. Nature, 618(7965):500–505, 2023.

[77] A Yu Kitaev. Quantum measurements and the abelian stabilizer problem. arXiv preprint

quant-ph/9511026, 1995.

[78] A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation. American

Mathematical Society, USA, 2002. ISBN 0821832298.

[79] Alexei Kitaev and William A. Webb. Wavefunction preparation and resampling using a

180

quantum computer, 2009.

[80] Natalie Klco and Martin J. Savage. Minimally entangled state preparation of localized wave
functions on quantum computers. Physical Review A, 102:012612, 2020. doi: 10.1103/
PhysRevA.102.012612.

[81] Rolf Landauer. Irreversibility and heat generation in the computing process. IBM Journal

of Research and Development, 5(3):183–191, 1961.

[82] Dean Lee, Jacob Watkins, Dillon Frame, Gabriel Given, Rongzheng He, Ning Li, Bing-Nan
Lu, and Avik Sarkar. Time fractals and discrete scale invariance with trapped ions. Physical
Review A, 100(1):011403, 2019.

[83] Dean Lee, Joey Bonitati, Gabriel Given, Caleb Hicks, Ning Li, Bing-Nan Lu, Abudit Rai,
Avik Sarkar, and Jacob Watkins. Projected cooling algorithm for quantum computation.
Physics Letters B, 807:135536, 2020.

[84]

Joonho Lee, Dominic W Berry, Craig Gidney, William J Huggins, Jarrod R McClean, Nathan
Wiebe, and Ryan Babbush. Even more efficient quantum computations of chemistry through
tensor hypercontraction. PRX Quantum, 2(3):030305, 2021.

[85] Xiaoyu Li, Guowu Yang, Carlos Manuel Torres Jr, Desheng Zheng, and Kang L Wang. A
class of efficient quantum incrementer gates for quantum circuit synthesis. International
Journal of Modern Physics B, 28(01):1350191, 2014.

[86] HQ Lin. Exact diagonalization of quantum-spin models. Physical Review B, 42(10):6561,

1990.

[87] Seth Lloyd. Universal quantum simulators. Science, 273(5278):1073–1078, 1996.

[88] Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by qubitization. Quantum, 3:

163, 2019.

[89] Guang Hao Low and Nathan Wiebe. Hamiltonian simulation in the interaction picture.

arXiv:1805.00675, 2018.

[90] Guang Hao Low, V. Kliuchnikov, and N. Wiebe. Well-conditioned multiproduct hamiltonian

simulation. arXiv: Quantum Physics, 2019.

[91] Shang-keng Ma. Introduction to the renormalization group. Reviews of Modern Physics, 45

(4):589, 1973.

[92] Pesi R Masani. Norbert Wiener 1894–1964, volume 5. Birkhäuser, 2012.

[93]

John C Mason and David C Handscomb. Chebyshev Polynomials. CRC press, 2002.

181

[94] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C Benjamin, and Xiao Yuan.
Quantum computational chemistry. Reviews of Modern Physics, 92(1):015003, 2020.

[95]

Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven.
Barren plateaus in quantum neural network training landscapes. Nature Communications, 9
(1):4812, 2018.

[96] Richard Meister and Simon C Benjamin. Resource-frugal hamiltonian eigenstate preparation

via repeated quantum phase estimation measurements. arXiv:2212.00846, 2022.

[97] Barry Simon Michael Reed. Methods of Modern Mathematical Physics, volume 1. Academic

Press, 1980.

[98] Christopher Monroe and Jungsang Kim. Scaling the ion trap quantum processor. Science,

339(6124):1164–1169, 2013.

[99] Chandra Sekhar Mukherjee, Subhamoy Maitra, Vineet Gaurav, and Dibyendu Roy. Preparing
dicke states on a quantum computer. IEEE Transactions on Quantum Engineering, 1:1–17,
2020.

[100] Pascal Naidon and Shimpei Endo. Efimov physics: a review. Reports on Progress in Physics,

80(5):056001, 2017.

[101] Michael A Nielsen and Isaac Chuang. Quantum computation and quantum information,

2002.

[102] Caleb J O’Loan.

Iterative phase estimation. Journal of Physics A: Mathematical and

Theoretical, 43(1):015301, 2009.

[103] Uri Peskin and Nimrod Moiseyev. The solution of the time-dependent schrödinger equation
by the (t, t’) method: Theory, computational algorithm and applications. The Journal of
Chemical Physics, 99(6):4590–4596, 1993.

[104] Diego Porras and J Ignacio Cirac. Effective quantum spin systems with trapped ions. Physical

Review Letters, 92(20):207901, 2004.

[105] David Poulin, Angie Qarry, Rolando Somma, and Frank Verstraete. Quantum simulation of
time-dependent hamiltonians and the convenient illusion of hilbert space. Physical Review
Letters, 106(17):170501, 2011.

[106] Zhengrong Qian, Jacob Watkins, Gabriel Given, Joey Bonitati, Kenneth Choi, and Dean Lee.

Demonstration of the rodeo algorithm on a quantum computer. arXiv:2110.07747, 2021.

[107] Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical Mathematics, volume 37.

Springer Science & Business Media, 2010.

182

[108] Yihui Quek, Daniel Stilck França, Sumeet Khatri, Johannes Jakob Meyer, and Jens Eisert.
Exponentially tighter bounds on limitations of quantum error mitigation. arXiv:2210.11505,
2022.

[109] Arthur G. Rattew and Bálint Koczor. Preparing arbitrary continuous functions in quantum

registers with logarithmic complexity, 2022.

[110] Arthur G Rattew, Yue Sun, Pierre Minssen, and Marco Pistoia. The efficient preparation of

normal distributions in quantum registers. Quantum, 5:609, 2021.

[111] Robert Raussendorf, Daniel E Browne, and Hans J Briegel. Measurement-based quantum

computation on cluster states. Physical Review A, 68(2):022312, 2003.

[112] Markus Reiher, Nathan Wiebe, Krysta M Svore, Dave Wecker, and Matthias Troyer. Eluci-
dating reaction mechanisms on quantum computers. Proceedings of the National Academy
of Sciences, 114(29):7555–7560, 2017.

[113] Gumaro Rendon, Jacob Watkins, and Nathan Wiebe. Improved accuracy for trotter simula-

tions using chebyshev interpolation. Quantum, 8:1266, 2024.

[114] T.J. Rivlin. Chebyshev Polynomials. Dover Books on Mathematics. Dover Publications,
2020. ISBN 9780486842332. URL https://books.google.com/books?id=3s0mygEACAAJ.

[115] Ankur Sarker, M Shamiul Amin, Avishek Bose, and Nafisah Islam. An optimized design
of binary comparator circuit in quantum computing. In 2014 International Conference on
Informatics, Electronics & Vision (ICIEV), pages 1–5. IEEE, 2014.

[116] Nicolas PD Sawaya, Tim Menke, Thi Ha Kyaw, Sonika Johri, Alán Aspuru-Guzik, and
Gian Giacomo Guerreschi. Resource-efficient digital quantum simulation of d-level systems
for photonic, vibrational, and spin-s hamiltonians. npj Quantum Information, 6(1):49, 2020.

[117] M.L. Shetterly and L. Freeman. Hidden Figures. HarperCollins, 2018.

ISBN

9780062881885.

[118] Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In
Proceedings 35th Annual Symposium on Foundations of Computer Science, pages 124–134.
IEEE, 1994.

[119] Avram Sidi. The Richardson Extrapolation Process, page 21–41. Cambridge Monographs
on Applied and Computational Mathematics. Cambridge University Press, 2003. doi: 10.
1017/CBO9780511546815.003.

[120] Daniel R Simon. On the power of quantum computation. SIAM Journal on Computing, 26

(5):1474–1483, 1997.

183

[121] Ionel Stetcu, Alessandro Baroni, and Joseph Carlson. Projection algorithm for state prepa-

ration on quantum computers. Physical Review C, 108(3):L031306, 2023.

[122] Masuo Suzuki. General theory of fractal path integrals with applications to many-body

theories and statistical physics. Journal of Mathematical Physics, 32(2):400–407, 1991.

[123] Masuo Suzuki. General theory of higher-order decomposition of exponential operators and

symplectic integrators. Physics Letters A, 165(5-6):387–395, 1992.

[124] Masuo Suzuki. Methodology of analytic and computational studies on quantum systems.

Journal of Statistical Physics, 110(3):945–956, 2003.

[125] Krysta M. Svore, Matthew B. Hastings, and Michael H. Freedman. Faster phase estimation.
Quantum Information & Computation, 14(3-4):306–328, 2014. doi: 10.26421/QIC14.3-4-7.

[126] Lloyd N. Trefethen. Appendix A. Six Myths of Polynomial Interpolation and Quadrature,
pages 263–271. SIAM, 2019. doi: 10.1137/1.9781611975949.appa. URL https://epubs.
siam.org/doi/abs/10.1137/1.9781611975949.appa.

[127] Jean-Philippe Uzan. The fundamental constants and their variation: observational and

theoretical status. Reviews of Modern Physics, 75(2):403, 2003.

[128] Dyon van Vreumingen and Kareljan Schoutens. Adiabatic ground-state preparation of
fermionic many-body systems from a two-body perspective. Physical Review A, 108(6):
062603, 2023.

[129] Farrokh Vatan and Colin Williams. Optimal quantum circuits for general two-qubit gates.

Physical Review A, 69:032315, 2004. doi: 10.1103/PhysRevA.69.032315.

[130] Almudena Carrera Vazquez, Daniel J Egger, David Ochsner, and Stefan Woerner. Well-
conditioned multi-product formulas for hardware-friendly hamiltonian simulation. Quantum,
7:1067, 2023.

[131] Libor Veis and Jiří Pittner. Adiabatic state preparation study of methylene. The Journal of

Chemical Physics, 140(21), 2014.

[132] Marco Vianello and Federico Piazzon. Stability inequalities for lebesgue constants via
markov-like inequalities. Dolomites Research Notes on Approximation, 11(DRNA Volume
11.1):1–9, 2018.

[133] Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee. Time-dependent hamil-

tonian simulation using discrete clock constructions. arXiv:2203.11353, 2022.

[134] James D Watson, Jacob Bringewatt, Alexander F Shaw, Andrew M Childs, Alexey V Gor-
shkov, and Zohreh Davoudi. Quantum algorithms for simulating nuclear effective field

184

theories. arXiv:2312.05344, 2023.

[135] Nathan Wiebe and Nathan S Babcock. Improved error-scaling for adiabatic quantum evolu-

tions. New Journal of Physics, 14(1):013024, 2012.

[136] Nathan Wiebe and Chris Granade. Efficient bayesian phase estimation. Physical Review

Letters, 117(1):010503, 2016.

[137] Nathan Wiebe, Dominic Berry, Peter Høyer, and Barry C Sanders. Higher order decomposi-
tions of ordered operator exponentials. Journal of Physics A: Mathematical and Theoretical,
43(6):065203, 2010. doi: 10.1088/1751-8113/43/6/065203.

[138] Nathan Wiebe, Dominic W Berry, Peter Høyer, and Barry C Sanders. Simulating quantum
dynamics on a quantum computer. Journal of Physics A: Mathematical and Theoretical, 44
(44):445308, 2011.

[139] Changhao Yi and Elizabeth Crosson. Spectral analysis of product formulas for quantum
ISSN 2056-6387. doi: 10.1038/

simulation. npj Quantum Information, 8(1):37, 2022.
s41534-022-00548-w.

[140] Jiehang Zhang, Guido Pagano, Paul W Hess, Antonis Kyprianidis, Patrick Becker, Harvey
Kaplan, Alexey V Gorshkov, Z-X Gong, and Christopher Monroe. Observation of a many-
body dynamical phase transition with a 53-qubit quantum simulator. Nature, 551(7682):
601–604, 2017.

[141] Sergiy Zhuk, Niall Robertson, and Sergey Bravyi. Trotter error bounds and dynamic multi-

product formulas for hamiltonian simulation, 2023.

[142] Christa Zoufal, Ryan V Mishmash, Nitin Sharma, Niraj Kumar, Aashish Sheshadri, Amol
Deshmukh, Noelle Ibrahim, Julien Gacon, and Stefan Woerner. Variational quantum al-
gorithm for unconstrained black box binary optimization: Application to feature selection.
Quantum, 7:909, 2023.

[143] Wojciech Hubert Zurek. Decoherence, einselection, and the quantum origins of the classical.

Reviews of Modern Physics, 75(3):715, 2003.

[144] Karol Życzkowski, Paweł Horodecki, Anna Sanpera, and Maciej Lewenstein. Volume of the

set of separable states. Physical Review A, 58(2):883, 1998.

185