APPLICATIONS OF NOISY INTERMEDIATE-SCALE QUANTUM COMPUTING
TO MANY-BODY NUCLEAR PHYSICS
By
Benjamin Prescott Hall
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics—Doctor of Philosophy
Computational Mathematics, Science and Engineering—Dual Major
2022
ABSTRACT
Many-body nuclear physics is the bridge that takes us from the fundamental laws gov-
erning individual nucleons to understanding how groups of them interact to form the nuclei
that lie at the heart of all atoms - the building blocks of our universe. Many powerful
techniques of classical computation have been developed over the years in order to study
ever more complex nuclear systems. However, we seem to be approaching the limits of
such classical techniques as the complexity of many-body quantum systems grows expo-
nentially with system size. Yet, the recent development of quantum computers offers one
hope as they are predicted to provide a significant advantage over classical computers in
certain applications, such as the quantum many-body problem. In this thesis, we focus on
developing and applying algorithms to solve various many-body nuclear physics problems
that can be run on the near-term quantum computers of the current noisy intermediate-scale
quantum (NISQ) era. As these devices have small qubit counts and high noise levels, we
focus our algorithms on various many-body toy models in order to gain insight and build
a foundation upon which future algorithms will be built to tackle the intractable problems
of our time. First, we tailor current quantum algorithms to efficiently run on NISQ era
devices and apply them to three pairing models of many-body nuclear physics: the Lipkin
model, the Richardson pairing model, and collective neutrino oscillations. We estimate
the ground-state energy of the first two models and simulate the time evolution and char-
acterize the entanglement of the third. Then, we develop two novel algorithm to increase
the efficiency and applicability of current NISQ era algorithms: an algorithm to compress
circuit depth to allow for less noisy computation, and a variational method to prepare an
important class of quantum states called Dicke states. Error mitigation techniques used to
improve the accuracy of results are also discussed and employed. All together, this work
provides a road map to apply the quantum computers of tomorrow to solve what nuclear
phenomena mystify us today.
This thesis is dedicated to my wife, Katherine - my rock,
and to my parents, David and Caroline - my first teachers.
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ACKNOWLEDGEMENTS
This dissertation would not exist without the support and guidance of my parents. They
sparked a passion for learning in me from a young age and gave me every opportunity
necessary to pursue whatever interested me. They have always been there for me. To my
wife Katherine, thank you for being my rock. It’s so kind of you to always listen to me
ramble on about physics and quantum computing. Without your love and support, I would
not have made it this far. To my brother Evan, thank you for being there for me to relax
with when I needed to recharge. I must also thank my fellow physicist, Jacob Watkins for
being such a good friend and making graduate school so enjoyable. Finally, I must thank
Pepper, my dog, for getting me through the final push to finish this thesis.
On for the academic side, I would like to thank my advisor Morten Hjorth-Jensen for
his guidance and mentorship. He has been my biggest cheerleader throughout my graduate
career. He’s let me explore a wide range of topics that interest me throughout my graduate
studies. Whenever I felt stuck or unworthy, his positivity would pick me right back up.
I must also thank all those that mentored me during the various internships I completed
during graduate school. To Joseph Carlson, Alessandro Roggero, and Alessandro Baroni,
thank you for mentoring me through the publishing of my first paper. I learned a great
deal about the fundamentals of quantum computing and neutrino physics from you. And
thank you to Patrick Coles for founding and running the summer school at Los Alamos
that made that work possible. To Gaute Hagen and Thomas Papenbrock from Oak Ridge,
thank you for guiding me through the nuclear theory required to complete the section of
my thesis that sparked my interest in this whole topic. I’d like to thank the Department
of Energy’s Office of Science Graduate Student Research Program for connecting me with
v
Oak ridge. Next, let me thank Davide Venturelli and everyone at the NASA QuAIL and
NAMS team for taking me in and letting me get experience running circuits on a real
quantum computer (thank you Riggeti) for the first time. That internship made me realize
my passion for pure quantum computing and circuit design. Furthermore, to Alessandro
Lavato and Yuri Alexeev at Argonne, thank you for teaching me about classical machine
learning and helping me better understand my own research on nuclear pairing. Finally, let
me thank all of the secretaries that have helped me along the way, without whom nothing
would have ever gotten done: Thank you to Amanda Martinez and Ashlee Adams from
LANL, Ryan Smith and Samantha Summers from DOE SCGSR, Tristyn Acasio and Saba
Hussain from NASA, and Caity Hitchens and Tamra Lagerwall from Argonne. A special
thanks to Kim Crosslan at MSU who’s always been there to guide me through my years
here and Elizebeth Deliyski at FRIB for helping me navigate the wonderful organization
that is nuclear physics at Michigan State.
Finally, let me acknowledge all of the scientists that have come before me and discovered
the knowledge upon which this thesis builds.
vi
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF ALGORITHMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2 MANY-BODY NUCLEAR THEORY . . . . . . . . . . . . . . . . 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Many-Body Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Classical Computation Techniques . . . . . . . . . . . . . . . . . . . 17
CHAPTER 3 QUANTUM COMPUTING . . . . . . . . . . . . . . . . . . . . . 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Variational Quantum Eigensolver . . . . . . . . . . . . . . . . . . . . 42
3.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
CHAPTER 4 LIPKIN MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Quantum Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
CHAPTER 5 PAIRING MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Quantum Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
CHAPTER 6 COLLECTIVE NEUTRINO OSCILLATIONS . . . . . . . . . . . 115
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Connection to the Pairing Model . . . . . . . . . . . . . . . . . . . . 122
vii
6.4 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.5 Dynamics of entanglement . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 Error mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.7 Additional data for concurrence and entanglement entropy . . . . . . 147
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
CHAPTER 7 QUANTUM CIRCUIT SQUEEZING ALGORITHM . . . . . . . . 151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Maximal Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 Ricochet Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4 Entanglement swapping . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.5 Dicke Subspace Modification . . . . . . . . . . . . . . . . . . . . . . 160
7.6 Entanglement Swapping Recursion . . . . . . . . . . . . . . . . . . . 165
7.7 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.8 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.9 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
CHAPTER 8 VARIATIONAL PREPARATION OF DICKE STATES . . . . . . 186
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.3 Calculating a Cost Function . . . . . . . . . . . . . . . . . . . . . . . 191
8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
CHAPTER 9 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
APPENDIX A SU(2) COMMUTATION RELATIONS . . . . . . . . . . . . . . 207
APPENDIX B NORMALIZED HARTREE-FOCK ANSATZ . . . . . . . . . . . 209
APPENDIX C LIPKIN HAMILTONIAN FOR HARTREE-FOCK . . . . . . . . 211
APPENDIX D DICKE STATE LEMMAS . . . . . . . . . . . . . . . . . . . . . 213
APPENDIX E STATE-OVERLAP ALGORITHM . . . . . . . . . . . . . . . . . 215
APPENDIX F PAIR COMMUTATION RELATIONS . . . . . . . . . . . . . . . 217
viii
LIST OF TABLES
Table 3.1: Comparison of basis, number of operators, and locality the of Jordan-
Wigner, parity-basis, and Bravyi-Kitiav transformations. . . . . . . . . 48
ix
LIST OF FIGURES
Figure 3.1: Depiction of the state |𝜓i on the Bloch sphere [20]. . . . . . . . . . . 35
Figure 3.2: Schematic of the Variational Quantum Eigensolver. . . . . . . . . . . 44
Figure 4.1: Schematic of Lipkin Model. . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 4.2: The energy eigenvalues (𝐸) of the Lipkin model, are plotted for
various interaction strengths (𝑉). The level degeneracy Ω and particle
number 𝑁 are both four while the single-particle energy 𝜖 is one. The
different energies 𝑒 𝑘 for 𝑘 = 0, 1, ..., 6 are depicted by different colors,
labeled in the plot itself. The solid lines are the results of the FCI
method while the dots are the results of the symmetry method. . . . . 56
Figure 4.3: The energy eigenvalues (𝐸) of the Lipkin model, computed via the
symmetry method, are plotted for various interaction strengths (𝑉).
The level degeneracy Ω and particle number 𝑁 are both ten while
the single-particle energy 𝜖 is one. The solid and dashed lines cor-
respond to signature numbers 𝑟 = +1 and 𝑟 = −1, respectively. The
colors yellow, magenta, cyan, red, green, and blue correspond to
𝑗 = 0, 1, ..., 5, respectively. . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 4.4: The expectation value of the Lipkin Hamiltonian in the SU(2) co-
herent state ansatz (4.30) is plotted vs theta for various values of 𝜒
which are distinguished by different colors, labeled on the plot itself.
The black dots represent the minimum energy (4.38). . . . . . . . . . 63
Figure 4.5: The Hartree-Fock and exact energies of the Lipkin are plotted against
various values of 𝜒. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 4.6: The relative difference between the Hartree-Fock and exact energies
is plotted against various values of 𝜒. . . . . . . . . . . . . . . . . . . 65
Figure 4.7: Comparison of energies for the Lipkin model calculated through
direct diagonalization (Exact), Hartree-Fock (HF), and the variational
quantum eigensolver (VQE). . . . . . . . . . . . . . . . . . . . . . . 72
x
Figure 5.1: Example schematic of the paring model with 𝑃 = 4 energy levels and
𝑁 = 2 pairs of fermions. Shown are four energy levels with single-
particle energies 𝑑0 , 𝑑1 , 𝑑2 , 𝑑3 of which the bottom two are initially
filled by pairs of fermions. The dashed line represents the Fermi level
which divides the energy levels with single-particle energies 𝑑0 and
𝑑1 (the hole states) from those with 𝑑2 and 𝑑3 (the particle states). . . . 75
Figure 5.2: Pair-Goldstone contraction schematic. . . . . . . . . . . . . . . . . . 87
Figure 5.3: Pair-Goldstone diagrams for the pairing model. . . . . . . . . . . . . . 89
Figure 5.4: Linear qubit-connectivity schematic. . . . . . . . . . . . . . . . . . . 96
Figure 5.5: Circular qubit-connectivity schematic. . . . . . . . . . . . . . . . . . 96
Figure 5.6: a) Linear particle-hole swap network (lphsn) for a four-particle, five-
hole system. b) Circular particle-hole swap network for a four-
particle, five-hole system (cphsn). See Figure 5.7 for schematic
representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Figure 5.7: a) Schematic representation of the linear particle-hole swap network
(lphsn) for a four-particle, five-hole system. b) Schematic repre-
sentation of the circular particle-hole swap network (cphsn) for a
four-particle, five-hole system. Each circle represents a qubit and a
slot (particle/hole). The particles are labeled 𝑝 0 , ..., 𝑝 3 and are col-
ored blue while the holes are labeled ℎ0 , ..., ℎ4 and are colored red.
The first and last columns of circles are the initial and final positions
of the qubits/slots. A rectangle around a pair of circles ( 𝑝𝑖 , ℎ 𝑗 ) de-
𝑝
notes that the gate 𝑆 𝐴 ℎ 𝑖𝑗 has been applied between the corresponding
qubits. See Figure 5.6 for circuit representation. . . . . . . . . . . . . 98
Figure 5.8: Initial correlation energies for the pairing model with 𝑃 = 4 energy
levels and 𝑁 = 2 pairs of fermions are compared to E_true, the true
ground state correlation energy. E_calc_ia uses the initial parame-
ters informed by MBPT (5.96) and E_calc_rand uses random initial
parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xi
Figure 5.9: The number of iterations required to minimize the correlation energy
(averaged over 10 trials) are compared for the pairing model with
𝑃 = 4 energy levels and 𝑁 = 2 pairs of particles. itr_rand and itr_ia
are the number of iterations required to minimize the correlation
energy for random initial parameters and initial parameters informed
by MBPT (5.96), respectively. . . . . . . . . . . . . . . . . . . . . . . 102
Figure 5.10: VQE calculated ground state correlation energies compared between
the case of random initial parameterization (E_calc_rand) and the
case of MBPT informed (5.96) initial parameterization (E_calc_ia). . . 102
Figure 5.11: Pairing strength vs ground state correlation energy obtained through
exact diagonalization (blue line), the variational quantum eigensolver
(red dots) and pair coupled cluster doubles theory (green dots) for
the pairing model with 𝑃 = 4 energy levels, 𝑁 = 2 pairs, linearly
increasing singular particle energies 𝑑 𝑝 = 𝑝, and constant pairing
strength 𝑔𝑖𝑎 = 𝑔. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 5.12: Pairing strength vs ground state correlation energy obtained through
exact diagonalization (blue line), the variational quantum eigensolver
(red dots) and pair coupled cluster doubles (green dots) for the pairing
model with 𝑃 = 4 energy levels, 𝑁 = 2 pairs, linearly increasing
singular particle energies 𝑑 𝑝 = 𝑝, and constant separable pairing
strength 𝑔𝑖𝑎 = 𝑔(𝑖𝑎/(max(𝑖)(max(𝑎)). . . . . . . . . . . . . . . . . . 104
Figure 5.13: Comparison of correlation energies calculated using the iterative
quantum excited states algorithm and direct diagonalization for the
pairing model with 𝑃 = 4 energy levels, 𝑁 = 2 pairs, linearly increas-
ing singular particle energy 𝑑 𝑝 = 𝑝, and constant pairing strength
𝑔𝑖𝑎 = 𝑔. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 5.14: Plot of the overlap squared between the actual ground state and cor-
responding Dicke state for pairing models with various values of 𝑝
and 𝑛 = b 𝑝/2c over increasing values of constant pairing strength. . . . 110
Figure 5.15: Plot of VQE estimated versus exact correlation energies for the pair-
ing model with 𝑃 = 4 energy levels and 𝑁 = 2 pairs of fermions
using the multi-configuration ansatz. . . . . . . . . . . . . . . . . . . 112
xii
Figure 5.16: Plot of relative error between VQE estimated and exact correlation
energies for the pairing model with 𝑃 = 4 energy levels and 𝑁 = 2
pairs of fermions using the multi-configuration ansatz. . . . . . . . . . 112
Figure 6.1: Panel (a) shows the error in matrix 2-norm equation (6.52) of the
two approximations 𝑈1 and 𝑈2 described in the text. Panel (b) shows
the state fidelity and the right panels show results for the inversion
probability 𝑃inv (𝑡). Panel (c) is for neutrino one while panel (d) is
for neutrino 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Figure 6.2: Pictorial representation of the swap network used in our simulation
in the case of 𝑁 = 4 neutrinos. . . . . . . . . . . . . . . . . . . . . . 131
Figure 6.3: Layout of the IBM Quantum Canary Processor Vigo [45]. Shown are
the five qubits, labeled from 0 to 4, and their connectivity denoted as
solid black lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Figure 6.4: Inversion probability 𝑃inv(𝑡) for neutrinos one and four: the red circle
and brown square correspond to the bare results, the blue triangle
and the green diamond are obtained after error mitigation (see text).
The left panel (VM) are virtual machine results while the right panel
(QPU) are results obtained on the Vigo [45] quantum device. . . . . . 134
Figure 6.5: Inversion probability 𝑃inv (𝑡) for neutrinos two and three. The notation
is the same as for Figure 6.4. . . . . . . . . . . . . . . . . . . . . . . 134
Figure 6.6: Inversion probability 𝑃inv at the initial time 𝑡 = 0 for the first neutrino.
Black solid circles are results from the Vigo QPU [45] while the red
squares correspond to results obtained using the VM with simulated
noise. Also shown are extrapolations to the zero noise limit, for both
the QPU (green line) and the VM (blue line), together with the extrap-
olated value (greed triangle up and blue triangle down respectively).
The dashed orange line denotes the result for a maximally mixed state. 135
xiii
Figure 6.7: Single spin entanglement entropy for neutrino 2. Black squares are
bare results obtained from the QPU, red triangles are results obtained
by amplifying the noise to 𝜖/𝜖0 = 3, the blue circles are obtained
using Richardson extrapolation, the turquoise plus symbols indicate
results obtained using the standard exponential extrapolation and the
green diamonds correspond to the results obtained from a shifted
exponential extrapolation using the maximum value of the entropy
(indicated as a dashed orange line). . . . . . . . . . . . . . . . . . . . 138
Figure 6.8: Pair entanglement entropy for the neutrino pair (1, 2) starting as
|𝑒i ⊗ |𝑒i (left panel) and pair (2, 4) which starts as the flavor state
|𝑒i ⊗ |𝑥i (right panel). Results obtained directly from the QPU are
shown as black squares (𝑟 = 1) and red triangles (𝑟 = 3) while blue
circles and green diamonds indicate mitigated results using Richard-
son and the shifted exponential extrapolations respectively. For the
shifted exponential ansatz we use the maximum value of the entropy
(indicated as a dashed orange line).The magenta triangle indicates
a mitigated result with shifted exponential extrapolation below zero
within errorbars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Figure 6.9: Extended concurrence 𝐶 e for two pairs of neutrinos, (1, 2) in the left
and (2, 4) in the right panel. The convention for the curves and date
point used here is the same as in Figure 6.8. The gray area indicates
the region where the concurrence 𝐶 (𝜌) is zero. The maximum value
for the concurrence is shown as a dashed orange line. . . . . . . . . . 141
Figure 6.10: Single spin entanglement entropy for all four neutrinos. Black squares
are bare results obtained from the QPU, the blue circles are obtained
using Richardson extrapolation and the green diamonds correspond
to the results obtained from a shifted exponential extrapolation using
the maximum value of the entropy (dashed orange line). The ma-
genta triangle indicates a mitigated result with shifted exponential
extrapolation below zero within errorbars. . . . . . . . . . . . . . . . 148
xiv
Figure 6.11: Pair entanglement entropy for all pairs of neutrinos. Black squares are
bare results obtained from the QPU, red triangles are results obtained
by amplifying the noise to 𝜖/𝜖0 = 3, the blue circles are obtained
using Richardson extrapolation and the green diamonds correspond
to the results obtained from a shifted exponential extrapolation using
the maximum value of the entropy (indicated as a dashed orange
line). The magenta triangle points are mitigated results with shifted
exponential extrapolation below zero within errorbars. . . . . . . . . . 148
Figure 6.12: Entanglement concurrence for all the pairs of qubits. The maximum
value for the concurrence is shown as a dashed orange line. . . . . . . 149
Figure 7.1: Graph of grid qubit-connectivity. Circled numbers represent qubits,
black lines represent that two qubits are connected, and black rectan-
gles represent gates 𝐴. . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Figure 7.2: Comparison of fidelities of estimated and exact density matrices of
both original and squeezed circuits. . . . . . . . . . . . . . . . . . . . 184
Figure 8.1: Overlap of deterministically and variationally prepared Dicke-states
with true Dicke-state. . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Figure 8.2: Comparison of deterministic and variational methods to prepare the
Dicke state |𝐷 42 i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
xv
LIST OF ALGORITHMS
Algorithm 5.1 Linear Particle-Hole Swap Network . . . . . . . . . . . . . . . . 99
Algorithm 5.2 Circular Particle-Hole Swap Network . . . . . . . . . . . . . . . . 100
Algorithm 7.1 Quantum Circuit Squeezing Algorithm . . . . . . . . . . . . . . . 175
Algorithm 8.1 Variational Mixing Algorithm . . . . . . . . . . . . . . . . . . . 191
xvi
CHAPTER 1
INTRODUCTION
Classical computation seems to be approaching the limit of its ability to simulate many-
body nuclear systems. Such systems are crucial to the understanding of nuclei, the heart of
every atom and thus one of the fundamental building blocks of our universe. The Hilbert
spaces of such systems grow exponentially with the number of particles in the system.
This often necessitates exponential growth of the computational complexity required to
solve such systems on a classical computer. Even modern methods, such as quantum
Monte Carlo, are hindered by problems, in this case, the fermion sign problem. The recent
development of quantum computers promises a plausible path forward to overcoming these
hurtles. Fault-tolerant quantum computers with a large enough number of qubits could
simulate large many-body quantum systems with ease. For example, Shor’s algorithm [75]
has been proven to be able to find the eigenenergies of a Hamiltonian with polynomial
complexity and time-evolution quantum algorithms benefit from a linear increase in qubits
with system size and access to quantum gates that simulate such evolution naturally.
However, today’s quantum computers are in what has been called the noisy intermediate-
scale quantum (NISQ) era, meaning that they have a small number of qubits (order 100)
and suffer from substantial noise due to qubit decoherence and measurement errors. Thus,
a large part of quantum computing research is currently focused on developing so called
near-term algorithms which can run on NISQ era quantum devices. Such algorithms are
often hybrid, meaning that they rely on both a quantum and a classical computer, and
variational, meaning that they are non-deterministic and rely on optimization algorithms
as a sub-routine. One such algorithm, the variational quantum eigensolver, has been used
1
to solve for the ground state energy of molecules as large as BeH2 [49] using six qubits
and hydrogen chains as large as H12 [64] with twelve qubits and 72 two-qubit gates. On the
nuclear physics side, the binding energy of the deuteron has been calculated on a quantum
computer [28]. In order to scale the applications of such algorithms to larger systems
which we cannot currently simulate classically, we will need to develope techniques that
allow for short depth circuit implementation on devices with limited qubit connectivity.
Furthermore, such algorithms will need to take advantage of specifics of the problem it is
simulating such as its symmetries and known approximations. Such is the central aim of
this thesis.
In this work, we develope and apply near-term algorithms to toy models of many-body
nuclear physics. The algorithms are used to estimate energy eigenvalues of the systems,
simulate time-evolution, and characterize entanglement. The techniques developed here
serve as a foundation upon which future algorithms can be developed to solve larger systems
whose simulation evades current classical computation. This thesis is organized as follows.
In chapters 2 and 3, we provide the background theory upon which the rest of the thesis
rests. Chapter 2 gives an overview of many-body nuclear theory, starting with quantum
mechanics itself, then going through the formalism used to describe many-body systems.
It ends with a review of classical computational techniques that are used to solve many-
body quantum problems. Chapter 3 gives an introduction to quantum computing including
its basic building blocks: qubits, gates, and circuits. It then gives an explanation of the
variational quantum eigensolver (VQE) and an overview of how to map problems from
fermionic space to the spin space required for quantum computation.
Chapters 4 through 6 are the heart of the thesis in that they develope techniques and
2
apply near-term algorithms to three toy pairing models of quantum many-body nuclear
physics. Chapter 4 focuses on the Lipkin model. Various classical solutions to the model
are discussed before walking through potential ways to solve the model on a quantum
computer. While the Lipkin model has been modeled on a quantum computer before,
this thesis provides a novel, short-depth ansatz. In chapter 5, we turn our attention to
the pairing model, whose classical solutions are discussed and against which quantum
solutions are bench-marked. The model is mapped to quantum gates and a novel ansatz
is developed for shortening the depth of the quantum circuit on a quantum computer with
circular qubit connectivity. A novel algorithm for calculating the energies of excited states
is also developed along with a new ansatz for solving the model with a large pairing
strength. Various initializations of the ansatz are introduced and compared. In chapter 6,
we simulate the time evolution and characterize the entanglement of collective neutrino
oscillations. The Hamiltonian is partitioned and the ansatz is constructed in a clever way so
as to maximize the quality of the quantum computer’s results. Error mitigation techniques
are introduced and applied in order to depress the interference of noise in the calculations
of the quantum computer.
Next, chapters 7 and 8 introduce two novel quantum algorithms which can be generally
applied to various areas of quantum computing in order to improve results. Chapter 7 lays
out the quantum circuit squeezing algorithm (QCSA) which trades an increase in number
of qubits for a decrease in circuit depth, thus potentially decreasing the accumulation
of noise. Chapter 8 discusses a novel approach to prepare Dicke states variationally.
Finally, in chapter 9 we present our conclusions and perspective on potential future work
in applications of NISQ era algorithms to many-body nuclear physics.
3
CHAPTER 2
MANY-BODY NUCLEAR THEORY
2.1 Introduction
In this chapter, we introduce the underlying theory of many-body nuclear physics,
starting with the basics of quantum mechanics, upon which all else is built. We then
give an introduction of the formalisms required to represent many-body systems, including
second quantization and particle-hole formalism. Finally, we describe various classical
computation techniques that are used to solve such many-body systems.
2.2 Quantum Mechanics
2.2.1 Introduction
To understand many-body nuclear theory, we must first understand the theory underling
all of nuclear physics: quantum mechanics. While the theory, like many physical theories,
has no exact beginning, one of its formalisms was put forth by the 1920s by Erwin
Schrödinger [73]. It is the fundamental theory upon which all other physics is built, setting
forth the laws that physical phenomena must follow at the smallest of scales. Quantum
mechanics, unlike classical mechanics (the set of laws that physical phenomena follow
at large scales) is inherently probabilistic. This means that the equations of quantum
mechanics can only give us the probability distribution for the possible values of physical
observables not predict, with certainty, what values will be measured. To thoroughly
introduce the theory, we begin with the postulates of the theory.
4
2.2.2 Postulates of Quantum Mechanics
Quantum mechanics is built on a set of postulates from which the rest of the theory is
constructed. The postulates are as follows:
Postulate 1: The state of a physical system is represented by a quantum state |Ψi which
belongs to a Hilbert space 𝐻.
Postulate 2: Every observable physical quantity A is represented by a Hermitian
operator 𝐴 acting in 𝐻.
Postulate 3: The result of measuring an observable A must be one of the eigenvalues
of its corresponding operator 𝐴.
Postulate 4: When an observable A is measured in the state |Ψi, the probability of
obtaining the eigenvalue 𝑎 𝑛 of 𝐴 is given by the squared amplitude of its corresponding
eigen-vector |Φ𝑛 i. One can expand the state |Ψi in terms of |Φ𝑛 i, the eigenvectors of 𝐴,
as follows
Õ
|Ψi = 𝑐 𝑛 |Φ𝑛 i . (2.1)
𝑛
Thus 𝑃(𝑎 𝑛 ), the probability of obtaining the eigenvalue 𝑎 𝑛 can be determined by
𝑃(𝑎 𝑛 ) = |𝑐 𝑛 | 2 = |hΦ𝑛 |Ψi| 2 . (2.2)
Postulate 5: If the measurement of an observable A results in 𝑎 𝑛 then the state
collapses to the normalized projection of |Ψi onto the eigen-subspace associated with 𝑎 𝑛
𝑃𝑛 |Ψi
Ψ→ p , (2.3)
hΨ|𝑃𝑛 |Ψi
where
Õ
𝑃𝑛 = |Φ𝑚 i hΦ𝑚 | , (2.4)
𝑚
5
and 𝑚 sums over the eigenvectors that correspond to the eigenvector 𝑎 𝑛 .
Postulate 6: The state |Ψi evolves in time according to the Schrödinger equation
𝑑
𝑖~ |Ψ(𝑡)i = 𝐻 |Ψ(𝑡)i . (2.5)
𝑑𝑡
2.2.3 The Schrödinger Equation
It is often useful to think about the time-independent and time-dependent parts of a
system separately. To separate the Schrödinger equation into time-dependent and time-
independent components, we assume that the state |Ψ(𝑥, 𝑡)i can be written as the product
of a time-independent |𝑋 (𝑥)i and a time-dependent |𝑇 (𝑡)i state:
|Ψ(𝑥, 𝑡)i = |𝑋 (𝑥)i |𝑇 (𝑡)i . (2.6)
The sixth postulate of quantum mechanics tells us that the state |Ψi of a system evolves in
time according to the Schrödinger equation (2.5). Plugging the separation (2.6) into the
Schrödinger equation (2.5) allows us to separate the differential equation:
𝑑
𝑖~ |Ψ(𝑥, 𝑡)i = 𝐻 |Ψ(𝑥, 𝑡)i , (2.7)
𝑑𝑡
becomes
𝑑
𝑖~ |𝑋 (𝑥)i |𝑇 (𝑡)i = 𝐻 |𝑋 (𝑥)i |𝑇 (𝑡)i , (2.8)
𝑑𝑡
which allows
1 𝑑 1
𝑖~ |𝑇 (𝑡)i = 𝐻 |𝑋 (𝑥)i = 𝐸, (2.9)
|𝑇 (𝑡)i 𝑑𝑡 |𝑋 (𝑥)i
6
where we were able to set both sides of (2.9) equal to the constant 𝐸 because they each
exclusively depend on different parameters, namely position 𝑥 an time 𝑡. From this, we
arrive at the time-independent Schrödinger equation (2.10) and time-evolution equation
(2.11) given below
𝐻 |𝑋 (𝑥)i = 𝐸 |𝑋 (𝑥)i , (2.10)
𝑑
𝑖~ |𝑇 (𝑡)i = 𝐸 |𝑇 (𝑡)i . (2.11)
𝑑𝑡
Solving the time evolution equation (2.11) yields the equation for the time-dependent state
|𝑇 (𝑡)i = 𝑒 −𝑖𝐸𝑡 , (2.12)
where we’ve set ~ = 1. This implies that the full state is given by
|Ψ(𝑥, 𝑡)i = 𝑒 −𝑖𝐸𝑡 𝑋 (𝑥)
∞
Õ (−𝑖𝑡) 𝑛 𝑛
= 𝐸 𝑋 (𝑥)
𝑛=0
𝑛!
∞
Õ (−𝑖𝑡) 𝑛 𝑛
= 𝐻 𝑋 (𝑥)
𝑛=0
𝑛!
= 𝑈 (𝑡) 𝑋 (𝑥), (2.13)
where we’ve defined the time-evolution operator 𝑈 (𝑡) as
𝑈 (𝑡) = 𝑒 −𝑖𝐻𝑡 . (2.14)
Here we’ve both assumed that 𝐻 is time-independent and used the time-independent
Schrödinger equation (2.10).
7
2.3 Many-Body Formalism
2.3.1 Introduction
Systems with multiple particles in quantum mechanics are called many-body systems.
The formalism describing such systems was developed most notably by Vladimir Fock [34].
In this formalism, which is called second-quantization, one keeps track of the number of
particles contained in each state. It is a powerful tool for understanding large systems of
particles in quantum mechanics.
2.3.2 Single-particle states
We define the coordinate representation of a single-particle state
𝜓 𝑝 (𝑥) = h𝑥| 𝑝i , (2.15)
to mean that the particle labeled 𝑝 is occupying the state which is characterized by the
set of coordinates 𝑥. Consider a complete set of orthonormal, single-particle states 𝑃 =
{𝑝 1 , . . . , 𝑝 𝑛 }. The orthogonality of the set means that for any states 𝑝, 𝑞 ∈ 𝑃, their inner
product is equal to the Kronecker delta,
∫ ∫
𝛿 𝑝,𝑞 = h𝑝|𝑞i = 𝑑𝑥 h𝑝|𝑥i h𝑥|𝑞i = 𝑑𝑥𝜓 ∗𝑝 (𝑥)𝜓 𝑞 (𝑥). (2.16)
The completeness of the set means that any single particle state can be expanded as a linear
combination of the set
Õ
|𝜓i = 𝑐 𝑝 | 𝑝i , (2.17)
𝑝
where
∫
𝑐 𝑝 = h𝑝|𝜓i = 𝑑𝑥𝜓 ∗𝑝 (𝑥)𝜓(𝑥). (2.18)
8
2.3.3 Multi-particle states
We now wish to consider a system of multiple identical fermions, each of which lives
in a Hilbert space 𝐻. Such a multi-particle systems lives in the Fock space 𝐹 (𝐻) which
is the direct sum of the anti-symmetric tensor product of the single-particle Hilbert spaces
𝐻,
Ê ∞
𝐹 (𝐻) = 𝑆− 𝐻 ⊗𝑁 , (2.19)
𝑁=0
where 𝑆− anti-symmetrizes tensors. Each tensor product of single-particle Hilbert spaces
𝐻 ⊗𝑁 = 𝐻 ⊗ . . . ⊗ 𝐻 contains multi-particle states
| {z }
𝑁 times
Ì 𝑁
𝜓 𝑝𝑖 (𝑥𝑖 ) = h𝑥 1 · · · 𝑥 𝑁 | 𝑝 1 · · · 𝑝 𝑁 i = | 𝑝 1 · · · 𝑝 𝑁 i , (2.20)
𝑖=1
Ë𝑁
where | 𝑝 1 · · · 𝑝 𝑛 i = 𝑖=1 | 𝑝𝑖 i. As indicated above, h𝑥1 · · · 𝑥 𝑁 | is often dropped, and
assumed to always be in order (1, . . . , 𝑁). Thus, if particles are permuted according to
a permutation matrix 𝜎, the state | 𝑝 𝜎1 · · · 𝑝 𝜎𝑁 i = h𝑥 1 · · · 𝑥 𝑁 | 𝑝 𝜎1 · · · 𝑝 𝜎𝑁 i implies that
particle labeled 𝑝 𝜎𝑖 is in the state characterized by the set of coordinates 𝑥𝑖 . The spaces
𝐻 ⊗𝑁 are anti-symmetrized because, according to the spin-statistics theorem, fermionic
wave-functions must be anti-symmetric with respect to the exchange of particles; that is
· · · 𝑝 𝑗 · · · 𝑝𝑖 · · · = − · · · 𝑝𝑖 · · · 𝑝 𝑗 · · · , (2.21)
which says that if particles 𝑝𝑖 and 𝑝 𝑗 swap states, the corresponding many-body state picks
up a minus sign. In general, a multi-particle fermionic state can pick up a factor of a
negative one depending on the permutation of the particles. That is
𝑝 𝜎1 . . . 𝑝 𝜎𝑁 = (−1) 𝜎 | 𝑝 1 · · · 𝑝 𝑁 i , (2.22)
9
where
+1
if 𝜎 is an even permutation
(−1) 𝜎 = (2.23)
−1
if 𝜎 is an odd permutation.
Thus, each anti-symmetric tensor product of single-particle Hilbert spaces 𝑆− 𝐻 ⊗𝑁 contains
anti-symmetrized multi-particle states
√
|{𝑝 1 . . . 𝑝 𝑁 }i = 𝑁!𝐴 | 𝑝 1 . . . 𝑝 𝑁 i , (2.24)
where 𝐴 is called the anti-symmetrizer and is defined as
1 Õ
𝐴= (−1) 𝜎 𝜎, (2.25)
𝑁! 𝜎∈𝑆
𝑁
and 𝑆 𝑁 is the symmetric group of order 𝑁. Applying the definition of the anti-symmetrizer
(2.25) to the definition of the anti-symmetrized multi-particle states (2.24) yields
1 Õ
|{𝑝 1 . . . 𝑝 𝑁 }i = √ (−1) 𝜎 𝜎 | 𝑝 1 . . . 𝑝 𝑁 i (2.26)
𝑁! 𝜎∈𝑆 𝑁
1 Õ
=√ (−1) 𝜎 | 𝑝 𝜎1 . . . 𝑝 𝜎𝑁 i (2.27)
𝑁! 𝜎∈𝑆 𝑁
1
= √ |𝑃|, (2.28)
𝑁!
where 𝑃 is an 𝑁 × 𝑁 matrix whose entries are equal to 𝑃𝑖 𝑗 = 𝜓 𝑝 𝑗 (𝑥𝑖 ); that is
𝜓 𝑝1 (𝑥 1 ) 𝜓 𝑝2 (𝑥 1 ) · · · 𝜓 𝑝 𝑁 (𝑥 1 )
1 𝜓 𝑝1 (𝑥 2 ) 𝜓 𝑝2 (𝑥 2 ) · · · 𝜓 𝑝 𝑁 (𝑥 2 )
|{𝑝 1 · · · 𝑝 𝑁 }i = √ .. .. .. . (2.29)
𝑁! . . .
𝜓 𝑝1 (𝑥 𝑁 ) 𝜓 𝑝2 (𝑥 𝑁 ) · · · 𝜓 𝑝 𝑁 (𝑥 𝑁 )
10
The above determinant is called a Slater determinant [77]. One can see that this state
is anti-symmetric with respect to the exchange of two particles as this corresponds to
exchanging two columns of the determinant, which picks up a factor of negative one. One
can see that the set of anti-symmetrized multi-particle states is orthonormal as, for any two
multi-particle state |{𝑝 1 · · · 𝑝 𝑛 }i , |{𝑞 1 · · · 𝑞 𝑛 }i ∈ 𝑆− 𝐻 ⊗𝑁 their overlap is
if 𝑝𝑖 = 𝑞𝑖 for all 𝑖 = 1, . . . , 𝑁
1 † 1 † 1
h{𝑝 1 · · · 𝑝 𝑛 }|{𝑞 1 · · · 𝑞 𝑛 }i = |𝑃 ||𝑄| = |𝑃 𝑄| =
𝑁! 𝑁!
0
otherwise,
(2.30)
as the entries of 𝑃† 𝑄 are given by
Õ𝑁 Õ 𝑁
†
(𝑃† 𝑄)𝑖 𝑗 = 𝑃𝑖𝑘 𝑄𝑘 𝑗 = 𝜓 ∗𝑝𝑖 (𝑥 𝑘 )𝜓 𝑞 𝑗 (𝑥 𝑘 ) = h𝑝𝑖 |𝑞 𝑗 i = 𝛿 𝑝𝑖 𝑞 𝑗 , (2.31)
𝑘 𝑘
which implies
Õ Ö𝑁
|𝑃† 𝑄| = (−1) 𝜎 𝛿 𝑝𝑖 𝑞 𝜎𝑖 . (2.32)
𝜎∈𝑆 𝑁 𝑖=1
Additionally, the set of anti-symmetrized multi-particle states is also complete. This means
that any anti-symmetrized multi-particle state can be expanded as
Õ
|𝜓i = 𝑐 𝑝1 ,...,𝑝 𝑁 |{𝑝 1 · · · 𝑝 𝑁 }i , (2.33)
𝑝 1 ,...,𝑝 𝑁
where the coefficients 𝑐 𝑝1 ,...,𝑝 𝑁 are given by
∫
𝑐 𝑝1 ,...,𝑝 𝑁 = h{𝑝 1 · · · 𝑝 𝑁 }|𝜓i = 𝑑𝑥1 . . . 𝑑𝑥 𝑛 𝜓 ∗𝑝1 (𝑥 1 ) . . . 𝜓 ∗𝑝 𝑁 (𝑥 𝑁 )𝜓(𝑥 1 , . . . , 𝑥 𝑁 ). (2.34)
2.3.4 Second Quantization
Second quantization (occupation number representation) is a formalism in which multi-
particle systems are described with the information of how many particles are in each state.
11
Its central ideas where first introduced by in 1927 Paul Dirac [24]. A fermionic multi-
particle state is described in second quantization as
|{𝑝 1 . . . 𝑝 𝑁 }i = |. . . 𝑛 𝑝1 . . . 𝑛 𝑝 𝑁 . . .i , (2.35)
where
if 𝑝𝑖 ∈ {𝑝 1 , . . . , 𝑝 𝑁 }
1
𝑛𝑖 =
0
if 𝑝𝑖 ∉ {𝑝 1 , . . . , 𝑝 𝑁 }.
Note that the occupation numbers 𝑛𝑖 can only be 0 or 1 as, according to the Pauli exclusion
principle, each state can contain at most one fermion. In this representation, it is convenient
to introduce creation (𝑎 † ) and annihilation (𝑎) operators which add and remove particles
from states, respectively. Acting on a fermionic single-particle state, they behave as follows:
𝑎 † |0i = |1i , 𝑎 † |1i = 0, (2.36)
𝑎𝑖 |0i = 0, 𝑎𝑖 |1i = |0i .
It can be seen that trying to increase the number of particles in a state above 1 or decrease the
number of particles in a state below 0 results in 0, reflecting the Pauli-exclusion principle.
Their behavior on multi-fermionic states is as follows:
𝑎𝑖† |𝑛1 . . . 𝑛𝑛 i = (−1) 𝑁𝑖 (1 − 𝑛𝑖 ) |𝑛1 . . . 𝑛𝑖−1 1𝑛𝑖+1 . . . 𝑛𝑛 i , (2.37)
𝑎𝑖 |𝑛1 . . . 𝑛𝑛 i = (−1) 𝑁𝑖 𝑛𝑖 |𝑛1 . . . 𝑛𝑖−1 0𝑛𝑖+1 . . . 𝑛𝑛 i , (2.38)
Í𝑖−1
where 𝑁𝑖 = 𝑗=1 𝑛 𝑗 . The (−1) 𝑁𝑖 factor comes from the anti-commutation relations of the
fermionic operators
n o
𝑎𝑖 , 𝑎 †𝑗 = 𝛿𝑖 𝑗 ,
12
𝑎𝑖 , 𝑎 𝑗 = {𝑎𝑖† , 𝑎 †𝑗 } = 0,
(2.39)
which, in turn, come from the fact that exchanging fermions picks up a factor of negative
one. Note that these operators can be used to create arbitrary fermionic multi-particle
states from the vacuum |0i as follows:
Ö 𝑁
𝑎 †𝑝𝑖 |0i = |{𝑝 1 · · · 𝑝 𝑁 }i . (2.40)
𝑖=1
One popular combination of these operators is the number operator
𝑁 𝑝 = 𝑎 †𝑝 𝑎 𝑝 , (2.41)
which counts the number of particles in a state (𝑝),
𝑁𝑝 · · · 𝑛𝑝 · · · = 𝑛𝑝 · · · 𝑛𝑝 · · · . (2.42)
These operators can also be used to represent arbitrary 𝑘-body operators in second
quantization as follows:
2 Õ
1
𝑂𝑘 = h𝑝 1 . . . 𝑝 𝑘 |𝑜 𝑘 |𝑞 1 . . . 𝑞 𝑘 i A 𝑎 †𝑝1 . . . 𝑎 †𝑝 𝑘 𝑎 𝑞 𝑘 . . . 𝑎 𝑞1 , (2.43)
𝑘! 𝑝 1 ...𝑝 𝑘
𝑞 1 ...𝑞 𝑘
where the anti-symmetrized 𝑘-body matrix element is defined as
h𝑝 1 . . . 𝑝 𝑘 |𝑜 𝑘 |𝑞 1 . . . 𝑞 𝑘 i 𝐴 = h{𝑝 1 . . . 𝑝 𝑘 }|𝑜 𝑘 |𝑞 1 . . . 𝑞 𝑘 i = h𝑝 1 . . . 𝑝 𝑘 |𝑜 𝑘 |{𝑞 1 . . . 𝑞 𝑘 }i ,
(2.44)
where
h𝑝 1 . . . 𝑝 𝑘 |𝑜 𝑘 |𝑞 1 . . . 𝑞 𝑘 i
∫
= 𝑑𝑥1 . . . 𝑑𝑥 𝑘 𝜓 ∗𝑝1 (𝑥 1 ) . . . 𝜓 ∗𝑝 𝑘 (𝑥 𝑘 )𝑜 𝑘 (𝑥 1 , . . . , 𝑥 𝑘 )𝜓 𝑞1 (𝑥 1 ) . . . 𝜓 𝑞 𝑘 (𝑥 𝑘 ). (2.45)
13
Note that from here on out in this thesis we will drop the brackets in Dirac-ket notation
when referring to anti-symmetrized states as we will be dealing exclusively with fermions
(whose states are always anti-symmetric). That is, from now on all |𝑥i should be taken to
mean |{𝑥}i.
2.3.5 Particle-Hole Formalism
When dealing with a large number of particles, it can become convenient to introduce
a reference state (|Φ0 i) other than the vacuum (|0i). To do this, we define a Fermi level
(𝐹) below which all states are assumed to be occupied and above which, all states are
assumed to be unoccupied. We call the states below the Fermi level hole states and those
above, particle states. The hole states are indexed with 𝑖, 𝑗, 𝑘, ... while the particle states
are indexed with 𝑎, 𝑏, 𝑐, .... If we wish to remain ambiguous as to whether or not a state is
a particle or hole state, we index with 𝑝, 𝑞, 𝑟, .... Thus, our reference state
|Φ0 i = |{𝑖1𝑖2 . . . 𝑖 𝑛 }i , (2.46)
in this case with 𝑛 hole states, acts as the new vacuum state. Because our new vacuum
already contains some particles, the notions of creation and annihilation must be redefined.
This is because, for example, annihilation of a particle that exists in the reference state
vacuum does not go to zero like it would if acting upon the true vacuum. Therefore, we
introduce particle-hole creation and annihilation operators
†
𝑎 𝑝
if 𝑝 > 𝐹 𝑏 𝑝
if 𝑝 > 𝐹
𝑏 †𝑝 = , 𝑏𝑝 = . (2.47)
†
𝑎 𝑝
if 𝑝 ≤ 𝐹
𝑏 𝑝
if 𝑝 ≤ 𝐹
In this way, applying a particle-hole creation operator (to the reference state) for a particle
in the reference state can be thought of as creating a hole (when, in reference to the true
14
vacuum, it destroys a particle), and vise verse.
2.3.6 Contractions
When calculating quantities with a large number of particles, it can become tedious to
manipulate the expression via successive applications of the fermionic anti-commutation
relations (2.39). One tool to alleviate this is the contraction. The contraction of two
operators 𝑋 and 𝑌 is defined as
𝑋𝑌 = 𝑋𝑌 − 𝑁 (𝑋𝑌 ), (2.48)
where 𝑁 (𝑋𝑌 ) indicates the normal order of 𝑋𝑌 . The normal order of a string of operators is
the one that vanishes when its expectation value is taken in the vacuum: h0| 𝑁 (𝑋𝑌 ) |0i = 0.
The normal order of pairs of the fermionic operators are
𝑁 (𝑎 𝑝 𝑎 𝑞 ) = 𝑎 𝑝 𝑎 𝑞 , (2.49)
𝑁 (𝑎 †𝑝 𝑎 †𝑞 ) = 𝑎 †𝑝 𝑎 †𝑞 , (2.50)
𝑁 (𝑎 †𝑝 𝑎 𝑞 ) = 𝑎 †𝑝 𝑎 𝑞 , (2.51)
𝑁 (𝑎 𝑝 𝑎 †𝑞 ) = −𝑎 †𝑞 𝑎 𝑝 . (2.52)
It follows that the contractions of the fermionic operators are
𝑎 𝑝 𝑎 𝑞 = 0, (2.53)
𝑎 †𝑝 𝑎 †𝑞 = 0, (2.54)
𝑎 †𝑝 𝑎 𝑞 = 0, (2.55)
𝑎 𝑝 𝑎 †𝑞 = 𝛿 𝑝𝑞 . (2.56)
15
In the particle-hole formalism, our normal orderings change as we have redefined the
vacuum: 𝑁 (𝑋𝑌 ) = hΦ0 | 𝑋𝑌 |Φ0 i. The normal order of pairs of particle-hole operators are
𝑁 (𝑎 𝑎 𝑎 𝑏 ) = 𝑎 𝑎 𝑎 𝑏 , 𝑁 (𝑎𝑖 𝑎 𝑗 ) = 𝑎𝑖 𝑎 𝑗 , (2.57)
𝑁 (𝑎 †𝑎 𝑎 †𝑏 ) = 𝑎 𝑎 𝑎 𝑏 , 𝑁 (𝑎𝑖† 𝑎 †𝑗 ) = 𝑎𝑖 𝑎 𝑗 , (2.58)
𝑁 (𝑎 †𝑎 𝑎 𝑏 ) = 𝑎 †𝑎 𝑎 𝑏 , 𝑁 (𝑎𝑖† 𝑎 𝑗 ) = −𝑎 †𝑗 𝑎𝑖 , (2.59)
𝑁 (𝑎 𝑎 𝑎 †𝑏 ) = −𝑎 †𝑏 𝑎 𝑎 , 𝑁 (𝑎𝑖 𝑎 †𝑗 ) = 𝑎𝑖† 𝑎 𝑗 . (2.60)
It follows that the contractions of the fermionic operators are
𝑎 𝑎 𝑎 𝑏 = 0, 𝑎𝑖 𝑎 𝑗 = 0, (2.61)
𝑎 †𝑎 𝑎 †𝑏 = 0, 𝑎𝑖† 𝑎 †𝑗 = 0, (2.62)
𝑎 †𝑎 𝑎 𝑏 = 0, 𝑎𝑖† 𝑎 𝑗 = 𝛿𝑖 𝑗 , (2.63)
𝑎 𝑎 𝑎 †𝑏 = 𝛿𝑎𝑏 , 𝑎𝑖 𝑎 †𝑗 = 0. (2.64)
One important theorem involving contractions is Wick’s theorem [86] which states that
a string of operators 𝐴𝐵𝐶𝐷... can be written as a sum of normal-ordered strings involving
all possible contractions; that is
Õ Õ
𝐴𝐵𝐶𝐷... = 𝑁 ( 𝐴𝐵𝐶𝐷...) + 𝑁 ( 𝐴𝐵𝐶𝐷...) + 𝑁 ( 𝐴𝐵𝐶𝐷...) + ..., (2.65)
singles doubles
where the sums over singles and doubles refer to all the possible ways the string of operators
can be contracted once and twice, respectively. Wick’s theorem is especially useful for
calculating expectation values as only the fully-contracted terms in the sum survive (all
other terms have leftover normal order terms which, by definition, have an expectation
value of zero). For example
h0| 𝐴𝐵𝐶𝐷 |0i = h0| 𝐴𝐵𝐶𝐷 |0i + h0| 𝐴𝐵𝐶𝐷 |0i . (2.66)
16
2.4 Classical Computation Techniques
2.4.1 Full Configuration Interaction
Consider the time-independent Schrodinger equation
𝐻 |𝜓i = 𝐸 |𝜓i . (2.67)
Expanding |𝜓i it in terms of an orthonormal basis {|𝑛i}
Õ
|𝜓i = 𝑐 𝑛 |𝑛i , (2.68)
𝑛
and inserting it into the Schrodinger equation (2.67), yields
Õ Õ
𝑐 𝑛 𝐻 |𝑛i = 𝐸 𝑐 𝑛 |𝑛i . (2.69)
𝑛 𝑛
Í
Inserting the identity 𝑘 |𝑘i h𝑘 | into the left-hand side yields
Õ Õ
𝐻 𝑘𝑛 𝑐 𝑛 |𝑘i = 𝐸 𝑐 𝑛 |𝑛i , (2.70)
𝑛 𝑛
where 𝐻 𝑘𝑛 = h𝑘 |𝐻|𝑛i. Left-multiplying both sides by h𝑚| yields
Õ
𝐻𝑚𝑛 𝑐 𝑛 = 𝐸𝑐 𝑚 , (2.71)
𝑛
which can be written in matrix form
𝐻𝐶 = 𝐸𝐶. (2.72)
Thus, diagonalizing 𝐻 will yield the eigenvalues 𝐸 and corresponding eigenfunctions
which are constructed from the coefficients 𝑐 𝑛 according to (2.68).
17
2.4.2 Symmetry Method
While full-configuration interaction theory can, in theory, be applied to any problem,
the trade-off is that, in order for this to be true, FCI is a problem agnostic theory. That is, it
does not take advantage of any of the information about the Hamiltonian itself. Without this
information, one is left to diagonalize a 22Ω × 22Ω matrix whose size grows exponentially
with the number of states Ω. However, if one incorporates some information about the
Hamiltonian, such as its symmetries, then one will only have to diagonalize several, small
matrices. The reason why will be seen in the following explanation of the symmetry
method.
In the symmetry method, one starts by identifying the symmetries 𝑆 𝑘 of the Hamilto-
nian, elements of the symmetry groups 𝑆 that commute with the Hamiltonian
[𝑆 𝑘 , 𝐻] = 0. (2.73)
The states are labeled by the quantum numbers corresponding to the irreducible represen-
tations of the symmetry groups. The Hamiltonian matrix is a block-diagonal matrix, with
one block for each irreducible representation. The proof is as follows: Let 𝑆 𝑘 be an element
of a symmetric group 𝑆 which commutes with the Hamiltonian. Also, let 𝑆 𝑘 |𝜓𝑛 i = 𝑠𝑛 |𝜓𝑛 i.
Then
1
h𝜓𝑛 |𝐻|𝜓𝑚 i = h𝜓𝑛 |𝐻𝑆 𝑘 |𝜓𝑚 i (2.74)
𝑠𝑚
1
= h𝜓𝑛 |𝑆 𝑘 𝐻|𝜓𝑚 i (2.75)
𝑠𝑚
𝑠𝑛
= h𝜓𝑛 |𝐻|𝜓𝑚 i , (2.76)
𝑠𝑚
which implies that (𝑠𝑚 − 𝑠𝑛 )𝐻𝑛𝑚 = 0. Thus, either 𝑠𝑛 = 𝑠𝑚 or 𝐻𝑛𝑚 = 0 and 𝑠𝑛 ≠ 𝑠𝑚 (|𝜓i 𝑛
and |𝜓i 𝑚 don’t have the same eigenvalue with respect to 𝑆 𝑘 ). Thus, the Hamiltonian matrix
18
is block diagonal.
2.4.3 Hartree-Fock Theory
Hartree-Fock theory attempts to approximate the ground state of a system by treating
it as a collection of non-interacting particles subject to a mean-field potential that ap-
proximates their interaction. In the theory, which was finalized in 1935 by Hartree [40],
one varies the single-particle orbital basis of a single Slater-determinant |Φi in order to
minimize the energy of that state; that is, the Hartree-Fock energy 𝐸 HF is defined as
hΦ|𝐻|Φi
𝐸 HF = min . (2.77)
|Φi hΦ|Φi
The variational principle guarantees that 𝐸 HF ≥ 𝐸 0 where 𝐸 0 is the ground state energy of
𝐻. Minimizing the functional will yield a set of equations which can be solved to determine
the Hartree-Fock Hamiltonian and its eigenvalues (single-particle energies) [55]. A small
variation in |Φi
|Φi → |Φi + |𝛿Φi , (2.78)
leads to the following variation in the energy
hΦ|𝐻|Φi hΦ|𝐻|Φ + 𝛿Φi
→ (2.79)
hΦ|Φi hΦ|Φ + 𝛿Φi
hΦ|𝐻|Φi hΦ|𝐻|𝛿Φi
= + (2.80)
hΦ|Φi + hΦ|𝛿Φi hΦ|Φi + hΦ|𝛿Φi
hΦ|𝐻|Φi hΦ|𝛿Φi hΦ|𝐻|𝛿Φi
= 1− + + O hΦ|𝛿Φi 2 (2.81)
hΦ|Φi hΦ|Φi hΦ|Φi
= 𝐸 + 𝛿𝐸, (2.82)
up to first order in hΦ|𝛿Φi, where
𝛿𝐸 = h𝛿Φ|𝐻 − 𝐸 |Φi , (2.83)
19
since 𝐻 |Φi = 𝐸 |Φi and hΦ|Φi = 1. The energy is stationary when
0 = 𝛿𝐸 = h𝛿Φ|𝐻 − 𝐸 |Φi . (2.84)
To make it so, we must first determine the form of |𝛿Φi. With
Ö 𝑛
|Φi = 𝑎𝑖† |0i , (2.85)
𝑖=1
infinitesimally varying the basis
Õ
𝑎𝑖† → e 𝑎𝑖† = (𝛿𝑖 𝑝 + 𝜖𝑖 𝑝 )𝑎 †𝑝 , (2.86)
𝑝
leads to the following variation in the wavefunction
Ö 𝑛
| Φi
e = 𝑎𝑖† |0i
e (2.87)
𝑖=1
𝑛
" #
Ö Õ
= (𝛿𝑖 𝑝 + 𝜖𝑖 𝑝 )𝑎 †𝑝 |0i (2.88)
𝑖=1 𝑝
𝑛
" #
Ö Õ
= 𝑎𝑖† + 𝜖𝑖 𝑝 𝑎 †𝑝 |0i (2.89)
𝑖=1 𝑝
Ö 𝑛 Õ © Ö 𝑛 𝑖−1
Ö
†ª † ª
𝑎𝑖† †©
= |0i +
𝑎 𝑗 ® 𝜖𝑖 𝑝 𝑎 𝑝 𝑎 𝑗 ® |0i + O (𝜖 2 ) (2.90)
𝑖=1 𝑝 « 𝑗=𝑖+1 « 𝑗=1 ¬
¬
= |Φi + |𝛿Φi , (2.91)
up to first order in 𝜖 where
Õ © Ö 𝑛 𝑖−1
Ö
† † †
|𝛿Φi = 𝑎 𝑗 ® |0i ,
ª © ª
𝑎 𝑗 ® 𝜖𝑖 𝑝 𝑎 𝑝 (2.92)
𝑝 « 𝑗=𝑖+1 « 𝑗=1 ¬
¬
which we can rewrite as
Õ © Ö 𝑛 Ö𝑖−1
†ª † †© † ª
|𝛿Φi =
𝑎 𝑗 ® 𝜖𝑖𝑎 𝑎 𝑎 𝑎𝑖 𝑎𝑖 𝑎 𝑗 ® |0i (2.93)
𝑎 « 𝑗=𝑖+1 « 𝑗=1 ¬
¬
20
Õ © 𝑛 † ª † ©Ö
Ö 𝑖−1
† ª
𝜖𝑖𝑎 𝑎 †𝑎 𝑎𝑖
=
𝑎 𝑗 ® 𝑎𝑖 𝑎 𝑗 ® |0i (2.94)
𝑎
¬ « 𝑗=1 ¬
𝑗=𝑖+1
«
Õ
= 𝜖𝑖𝑎 𝑎 †𝑎 𝑎𝑖 |Φi , (2.95)
𝑎
since 𝑝 must be a particle state in order for |𝛿Φi ≠ 0 and
𝑖−1
Ö 𝑖−1
Ö
𝑎𝑖 𝑎𝑖† 𝑎 †𝑗 ® |0i = ({𝑎𝑖 , 𝑎𝑖† } − 𝑎𝑖† 𝑎𝑖 ) 𝑎 †𝑗 ® |0i
© ª © ª
« 𝑗=1 ¬ « 𝑗=1 ¬
𝑖−1
©Ö † ª
= (1 − 𝑎𝑖† ) 𝑎 𝑗 ® 𝑎𝑖 |0i
« 𝑗=1 ¬
𝑖−1
©Ö † ª
= 𝑎 𝑗 ® |0i . (2.96)
« 𝑗=1 ¬
Having determined the form of |𝛿Φi, we can plug it into the condition that the energy is
stationary (2.84) to yield
0 = h𝛿Φ|𝐻 − 𝐸 |Φi (2.97)
Õ
= 𝜖𝑖𝑎 hΦ| 𝑎𝑖† 𝑎 𝑎 (𝐻 − 𝐸) |Φi (2.98)
𝑎
Õ
𝜖𝑖𝑎 hΦ| 𝑎𝑖† 𝑎 𝑎 𝐻 |Φi − 𝐸 𝑎𝑖†
= hΦ| 𝑎 𝑎 |Φi , (2.99)
𝑎
which holds if
0 = hΦ| 𝑎𝑖† 𝑎 𝑎 𝐻 |Φi (2.100)
Õ 1 Õ 𝑝𝑞
𝑡 𝑞 hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 𝑞 |Φi + 𝑣 𝑟 𝑠 hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 |Φi
𝑝
= (2.101)
𝑝𝑞
4 𝑝𝑞𝑟 𝑠
Õ
𝑡 𝑞 hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 𝑞 |Φi
𝑝
=
𝑝𝑞
"
1 Õ 𝑝𝑞
+ 𝑣 𝑟 𝑠 hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 |Φi + hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 |Φi
4 𝑝𝑞𝑟 𝑠
21
#
+ hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 |Φi + hΦ| 𝑎𝑖† 𝑎 𝑎 𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 |Φi (2.102)
Õ
𝑝 1 Õ 𝑝𝑞
= 𝑡 𝑞 𝛿 𝑝𝑎 𝛿 𝑞𝑖 + 𝑣 (ℎ(𝑠)𝛿 𝑝𝑎 𝛿 𝑞𝑠 𝛿𝑟𝑖 − ℎ(𝑠)𝛿 𝑝𝑠 𝛿 𝑞𝑎 𝛿𝑟𝑖
𝑝𝑞
4 𝑝𝑞𝑟 𝑠 𝑟 𝑠
+ ℎ(𝑟)𝛿 𝑝𝑎 𝛿 𝑞𝑟 𝛿 𝑠𝑖 − ℎ(𝑟)𝛿 𝑝𝑟 𝛿 𝑞𝑎 𝛿 𝑠𝑖 ) (2.103)
1 Õ 𝑎𝑗 𝑗𝑎 𝑎𝑗 𝑗𝑎
= 𝑡𝑖𝑎 + 𝑣𝑖 𝑗 − 𝑣 𝑖 𝑗 − 𝑣 𝑗𝑖 + 𝑣 𝑗𝑖 (2.104)
4 𝑗
Õ
𝑎𝑗
= 𝑡𝑖𝑎 + 𝑣𝑖 𝑗 , (2.105)
𝑗
which can be written as h𝑎|𝐻HF |𝑖i = 0 by defining the Hartree-Fock Hamiltonian
Õ
𝑝𝑗
𝐻HF = 𝑡 + 𝑣 𝑞𝑖 |𝑟i h𝑠| . (2.106)
𝑟 𝑠𝑖
The single-particle energies are the eigenvalues of the Hartree-Fock Hamiltonian
𝜖 𝑝 = h𝑝| 𝐻HF | 𝑝i (2.107)
Õ
𝑝 𝑝𝑖
= 𝑡𝑝 + 𝑣 𝑝𝑖 , (2.108)
𝑖
where we’ve relabeled 𝑗 ↔ 𝑖.
2.4.4 Many-Body Perturbation Theory
To go beyond Hartree-Fock, we introduce many-body permutation theory [52] which
adds in correlations between particles as perturbations to the Hartree-Fock wavefunction.
Analogous to single-body perturbation theory, it is possible to approximate the energy of
a system by writing the Hamiltonian as the sum of two terms, one of which is readily
diagonalizable. An introduction of this theory is given here as its results will be used to
inform our initialization of the variational quantum algorithms which will be used later.
Our aim is to solve the Schrödinger equation
𝐻 |Ψ0 i = 𝐸 |Ψ0 i . (2.109)
22
For perturbation theory, we assume that our Hamiltonian 𝐻 can be written as the sum of
an unperturbed Hamiltonian 𝐻0 and an interacting Hamiltonian 𝐻 𝐼
𝐻 = 𝐻0 + 𝐻 𝐼 , (2.110)
where the Schrödinger equation for 𝐻0
𝐻0 |Φ𝑛 i = 𝐸 𝑛 |Φ𝑛 i , (2.111)
is easily solvable. We expand the exact wave-function |Ψ0 i in terms of the unperturbed
wave-function
Õ∞
|Ψ0 i = |Φ0 i + 𝐶𝑛 |Φ𝑛 i . (2.112)
𝑛=1
Employing intermediate normalization hΦ0 |Ψ0 i = 1, along with equations (2.109) through
(2.111), we can derive
𝐸 = hΦ0 |𝐻|Ψ0 i = hΦ0 |𝐻0 |Ψ0 i + hΦ0 |𝐻 𝐼 |Ψ0 i = 𝐸 0 + hΦ0 |𝐻 𝐼 |Ψ0 i , (2.113)
from which we define
Δ𝐸 = 𝐸 − 𝐸 0 = hΦ0 |𝐻 𝐼 |Ψ0 i . (2.114)
We now introduce the following operators
𝑃 = |Φ0 i hΦ0 | , (2.115)
Õ ∞
𝑄= |Φ𝑛 i hΦ𝑛 | . (2.116)
𝑛=1
Noting that 𝑃 + 𝑄 = 1 and using the expanded form of the exact wave-function (2.112) we
can rewrite the exact wave-function as
|Ψ0 i = (𝑃 + 𝑄) |Ψ0 i = |Φ0 i + 𝑄 |Ψ0 i . (2.117)
23
To determine the second term, note that by plugging in the summed version of the Hamil-
tonian (2.110) into the exact Schrödinger equation (2.109), rearranging, and adding the
term 𝐸 0 |Ψ0 i (for Rayleigh-Schrodinger perturbation theory) to both sides, one arrives at
(𝐸 0 − 𝐻0 ) |Ψ0 i = (𝐻 𝐼 − Δ𝐸) |Ψ0 i , (2.118)
where we’ve used the definition of Δ𝐸 (2.114). Applying 𝑄 to both sides implies
𝑄
𝑄 |Ψ0 i = (𝐻 𝐼 − Δ𝐸) |Ψ0 i . (2.119)
𝐸 0 − 𝐻0
Plugging this into (2.117) yields
𝑄
|Ψ0 i = |Φ0 i + (𝐻 𝐼 − Δ𝐸) |Ψ0 i , (2.120)
𝐸 0 − 𝐻0
Inserting this equation into itself iteratively yields
∞ 𝑛
Õ 𝑄
|Ψ0 i = (𝐻 𝐼 − Δ𝐸) |Φ0 i , (2.121)
𝑛=0
𝐸 0 − 𝐻0
which, when plugged into the expression for the energy (2.114), yields
∞ 𝑛
Õ 𝑄
Δ𝐸 = hΦ0 | 𝐻 𝐼 (𝐻 𝐼 − Δ𝐸) |Φ0 i . (2.122)
𝑛=0
𝐸 0 − 𝐻0
Note that since 𝑄 commutes with 𝐻0 and Δ𝐸 is constant, we have that 𝑄Δ𝐸 |Φ0 i =
Δ𝐸𝑄 |Φ0 i = 0 and thus the energy becomes
∞ 𝑛−1
Õ 𝑄 𝑄
Δ𝐸 = hΦ0 |𝐻 𝐼 |Φ0 i + hΦ0 | 𝐻 𝐼 (𝐻 𝐼 − Δ𝐸) 𝐻 𝐼 |Φ0 i . (2.123)
𝑛=1
𝐸 0 − 𝐻0 𝐸 0 − 𝐻0
Perturbatively expanding in term of 𝐻 𝐼 , which is assumed to be small, we can write
Õ ∞
Δ𝐸 = Δ𝐸 (𝑛) . (2.124)
𝑛=1
24
Comparing (2.123) and (2.124), we identify
Δ𝐸 (1) = hΦ0 |𝐻 𝐼 |Φ0 i , (2.125)
𝑄
Δ𝐸 (2) = hΦ0 |𝐻 𝐼 𝐻 𝐼 |Φ0 i , (2.126)
𝐸 0 − 𝐻0
𝑄 𝑄
Δ𝐸 (3) = hΦ0 |𝐻 𝐼 𝐻𝐼 𝐻 𝐼 |Φ0 i , (2.127)
𝐸 0 − 𝐻0 𝐸 0 − 𝐻0
𝑄 𝑄
− hΦ0 |𝐻 𝐼 hΦ0 |𝐻 𝐼 |Φ0 i 𝐻 𝐼 |Φ0 i , (2.128)
𝐸 0 − 𝐻0 𝐸 0 − 𝐻0
where we’ve used (2.114) for the last expression. Recall that we assumed that our Hamil-
tonian
Õ 1 Õ 𝑝𝑞 † †
𝑡 𝑞 𝑎 †𝑝 𝑎 𝑞 +
𝑝
𝐻= 𝑣 𝑎 𝑎 𝑎 𝑠 𝑎𝑟 , (2.129)
𝑝𝑞
4 𝑝𝑞𝑟 𝑠 𝑟 𝑠 𝑝 𝑞
could be partitioned (2.110) as 𝐻 = 𝐻0 + 𝐻 𝐼 where 𝐻0 is easily solvable. There are
several ways to partition the Hamiltonian as such. The one we will consider here is called
Hartree-Fock partitioning [80] in which we set the unperturbed Hamiltonian 𝐻0 equal to
the Hartree-Fock Hamiltonian (2.106) since it can easily be diagonalized as
Õ
𝐻0 = 𝜖 𝑝 𝑎 †𝑝 𝑎 𝑝 , (2.130)
𝑝
𝑝 Í 𝑝𝑖
where 𝜖 𝑝 are the single-particle energies (2.107) which are given by 𝜖 𝑝 = 𝑡 𝑝 + 𝑖 𝑣 𝑝𝑖 . With
this partitioning, the interacting Hamiltonian 𝐻 𝐼 becomes
𝐻 𝐼 = 𝐻 − 𝐻0
Õ 1 Õ 𝑝𝑞 † †
(𝑡 𝑞 − 𝜖 𝑝 𝛿 𝑝𝑞 )𝑎 †𝑝 𝑎 𝑞 +
𝑝
= 𝑣 𝑎 𝑎 𝑎 𝑠 𝑎𝑟 . (2.131)
𝑝𝑞
4 𝑝𝑞𝑟 𝑠 𝑟 𝑠 𝑝 𝑞
Applying the unperturbed Hamiltonian 𝐻0 (2.130) to the reference state |Φ0 i yields
!
Õ Ö
† †
𝐻0 |Φ0 i = 𝜖𝑝𝑎𝑝𝑎𝑝 𝑎 𝑞 𝑛 |0i (2.132)
𝑝 𝑛
25
𝑚−1
! 𝑁
!
Õ Õ Ö Ö
𝑎 † 𝑎 𝑝 𝑎 †𝑞 𝑛 𝑎 †𝑞 𝑚 𝑎 †𝑞 𝑛 |0i
= 𝜖𝑝 𝑝 (2.133)
𝑝 𝑚 𝑛=1 𝑛=𝑚+1
!
Õ Õ Ö
= 𝜖𝑝 (−1) 𝑚−1 𝛿 𝑝𝑞 𝑚 𝑎 †𝑝 𝑎 †𝑞 𝑛 |0i (2.134)
𝑝 𝑚 𝑛
!
Õ Õ Ö
= 𝜖𝑝 𝛿 𝑝𝑞 𝑚 𝑎 †𝑞 𝑛 |0i (2.135)
𝑝 𝑚 𝑛
Õ
= 𝜖 𝑞 𝑚 |Φ0 i , (2.136)
𝑚
which implies that, with the notation 𝐻0 |Φ0 i = 𝐸 0 |Φ0 i, we have
Õ
𝐸0 = 𝜖𝑛 , (2.137)
𝑛
where we’ve relabeled 𝑚 → 𝑛. Using (2.136), we can see that applying 𝐻0 to excited
forms of the reference state yields
E E
𝐸𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
Φ𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
= 𝐻0 Φ𝑖𝑎11...𝑖 ...𝑎 𝑛
𝑛
(2.138)
𝑛
© Õ Õ
= 𝜖𝑞𝑛 + 𝜖 𝑎 𝑚 ® |0i
ª
(2.139)
«𝑞 𝑛 ∉{𝑖1 ,...,𝑖 𝑛 } 𝑚=1 ¬ !
Õ Õ 𝑛 Õ 𝑛
= 𝜖𝑞𝑛 + 𝜖𝑎𝑚 − 𝜖𝑖 𝑚 |0i (2.140)
𝑞𝑛 𝑚=1 𝑚=1
= 𝐸 0 − 𝜖𝑖𝑎11...𝑖 ...𝑎 𝑛
𝑛
|0i , (2.141)
which implies that
𝐸 0 − 𝐸𝑖𝑎11...𝑖...𝑎 𝑛
𝑛
= 𝜖𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
, (2.142)
where we’ve defined
Õ 𝑛
𝜖𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
= 𝜖𝑖 𝑚 − 𝜖 𝑎 𝑚 . (2.143)
𝑚=1
26
With Hartree-Fock partitioning, we can use (2.142) to write the correlation energies (2.125
- 2.127) more explicitly. Starting with Δ𝐸 (2) (2.126) we have
𝑄
Δ𝐸 (2) = hΦ0 | 𝐻 𝐼 𝐻 𝐼 |Φ0 i , (2.144)
𝐸 0 − 𝐻0
and noting that
𝑄
= 𝑄 (𝐸 0 − 𝐻0 ) −1 𝑄 (2.145)
𝐸 0 − 𝐻0
Õ
= |Φ𝑛 i hΦ𝑛 | (𝐸 0 − 𝐻0 ) −1 |Φ𝑚 i hΦ𝑚 | (2.146)
𝑛,𝑚=1
Õ
= 𝛿𝑛𝑚 |Φ𝑛 i (𝐸 0 − 𝐸 𝑚 ) −1 hΦ𝑚 | (2.147)
𝑛,𝑚=1
Õ
= |Φ𝑛 i (𝐸 0 − 𝐸 𝑛 ) −1 hΦ𝑛 | (2.148)
𝑛=1
since 𝑄 is idempotent, we have that
Õ hΦ0 |𝐻 𝐼 |Φ𝑎 i hΦ𝑎 |𝐻 𝐼 |Φ0 i Õ hΦ0 |𝐻 𝐼 |Φ𝑖𝑎𝑏𝑗 i hΦ𝑖𝑎𝑏𝑗 |𝐻 𝐼 |Φ0 i
𝑖 𝑖
𝑎 + 𝑎𝑏
, (2.149)
𝑎𝑖
𝜖 𝑖 𝑎<𝑏,𝑖< 𝑗
𝜖 𝑖𝑗
as any higher terms would be automatically zero as 𝐻 only includes up to two-body
interactions. The only non-zero contribution is
Õ
𝑣 𝑟 𝑠 hΦ0 | {𝑎𝑖† 𝑎 †𝑗 𝑎 𝑎 𝑎 𝑏 }{𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 } |Φ0 i
𝑝𝑞
hΦ𝑖𝑎𝑏𝑗 |𝐻 𝐼 |Φ0 i = (2.150)
𝑝𝑞
1 Õ 𝑝𝑞
= 𝑣 hΦ0 | {𝑎𝑖† 𝑎 †𝑗 𝑎 𝑎 𝑎 𝑏 }{𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 } |Φ0 i
4 𝑝𝑞 𝑟 𝑠
+ hΦ0 | {𝑎𝑖† 𝑎 †𝑗 𝑎 𝑎 𝑎 𝑏 }{𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 } |Φ0 i
+ hΦ0 | {𝑎𝑖† 𝑎 †𝑗 𝑎 𝑎 𝑎 𝑏 }{𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 } |Φ0 i
27
+ hΦ0 | {𝑎𝑖† 𝑎 †𝑗 𝑎 𝑎 𝑎 𝑏 }{𝑎 †𝑝 𝑎 †𝑞 𝑎 𝑠 𝑎𝑟 } |Φ0 i (2.151)
1 Õ 𝑝𝑞
= 𝑣 (𝛿𝑖𝑟 𝛿 𝑗 𝑠 𝛿𝑎𝑞 𝛿 𝑏 𝑝 − 𝛿𝑖𝑠 𝛿 𝑗𝑟 𝛿𝑎𝑞 𝛿 𝑏 𝑝
4 𝑝𝑞 𝑟 𝑠
− 𝛿𝑖𝑟 𝛿 𝑗 𝑠 𝛿𝑎 𝑝 𝛿 𝑏𝑞 + 𝛿𝑖𝑠 𝛿 𝑗𝑟 𝛿𝑎 𝑝 𝛿 𝑏𝑞 ) (2.152)
1 𝑏𝑎
= 𝑣 𝑖 𝑗 − 𝑣 𝑏𝑎
𝑗𝑖 − 𝑣 𝑎𝑏
𝑖𝑗 + 𝑣 𝑎𝑏
𝑗𝑖 (2.153)
4
= 𝑣 𝑎𝑏
𝑗𝑖 . (2.154)
𝑗𝑖
In an analogous manner, one can compute hΦ0 |𝐻 𝐼 |Φ𝑖𝑎𝑏𝑗 i = 𝑣 𝑎𝑏 which yields
Õ 𝑣 𝑖𝑎𝑏𝑗 𝑣 𝑖𝑎𝑏𝑗
(2)
Δ𝐸 = . (2.155)
𝑎<𝑏,𝑖< 𝑗
𝜖𝑖𝑎𝑏𝑗
The same procedure can be carried out for higher order terms [53]. For example, the third
order contribution to the correlation energy is given by
Õ 𝑣 𝑖 𝑗 𝑣 𝑏𝑘 𝑣 𝑎𝑐 𝑖𝑗
𝑣 𝑐𝑑 𝑣 𝑐𝑑 𝑎𝑏 Õ 𝑣 𝑎𝑏 𝑘𝑙 𝑖 𝑗
𝑎𝑏 𝑣 𝑖 𝑗 𝑘𝑙 𝑣 𝑖 𝑗 𝑣 𝑎𝑏
Õ
(3) 𝑎𝑏 𝑖𝑐 𝑖𝑘
Δ𝐸 = + + . (2.156)
𝑖< 𝑗 <𝑘
𝜖𝑖𝑎𝑏 𝑎𝑐
𝑗 𝜖𝑖𝑘 𝑖< 𝑗 𝜖𝑖𝑎𝑏 𝑐𝑑
𝑗 𝜖𝑖 𝑗 𝑖< 𝑗 <𝑘 <𝑙
𝜖𝑖𝑎𝑏 𝑎𝑏
𝑗 𝜖 𝑘𝑙
𝑎<𝑏<𝑐 𝑎<𝑏<𝑐<𝑑 𝑎<𝑏
2.4.5 Coupled Cluster Theory
Coupled cluster theory (CC) was first developed to study nuclear physics by Coester
and Kümmel [32] in the 1950’s. The ansatz for the theory is given by
|Ψi = 𝑒𝑇 |Φ0 i , (2.157)
where |Φ0 i is the reference state and 𝑇, the cluster operator, is defined as
Õ 𝐴
𝑇= 𝑇𝑛 , (2.158)
𝑝=𝑛
28
where 𝐴 is the maximum number of particle-hole excitations and each term in the sum is
given by
1 Õ 𝑎1 ...𝑎 𝑛 †
𝑇𝑛 = 𝑡 𝑎 . . . 𝑎 †𝑎 𝑛 𝑎𝑖 𝑛 . . . 𝑎𝑖1 . (2.159)
𝑛! 𝑖 ...𝑖 𝑖1 ...𝑖 𝑛 𝑎1
1 𝑛
𝑎 1 ...𝑎 𝑛
Often the cluster operator is truncated to a small number of terms. A common example
is to truncate to 𝑁 = 2 leading to 𝑇 = 𝑇1 + 𝑇2 ; this is the so-called singles and doubles
approximation (CCSD). Starting with the time-independent Schrödinger equation
𝐻 |Ψi = 𝐸 |Ψi , (2.160)
inserting the coupled cluster ansatz (2.157) and left-multiplying both sides by 𝑒 −𝑇 , and
then left-multiplying by either the reference state or an excited state yields
𝐸 = hΦ0 |𝐻|Φ0 i , (2.161)
0 = hΦ𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
|𝐻|Φ0 i , (2.162)
respectively. Here 𝐻¯ is the similarity transformed Hamiltonian
𝐻 = 𝑒 −𝑇 𝐻𝑒𝑇 . (2.163)
Additionally, the excited states notation is as follows
|Φ𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
i = 𝑎 †𝑎1 . . . 𝑎 †𝑎 𝑛 𝑎𝑖1 . . . 𝑎𝑖 𝑛 |Φ0 i . (2.164)
Finally, after defining the reference energy
𝐸 ref = hΦ|𝐻|Φi , (2.165)
29
we define the correlation energy (Δ𝐸) to be the difference between the CC energy 𝐸 as
defined in 2.161 and the reference energy
Δ𝐸 = 𝐸 − hΦ|𝐻|Φi , (2.166)
which subtracts of the reference energy. Note that the energy (2.161) depends on the cluster
amplitudes 𝑡𝑖𝑎11...𝑖
...𝑎 𝑛
𝑛
, which can be obtained deterministically by solving the amplitude equa-
tions (2.162). In order to do so, one often expands the similarity transformed Hamiltonian
𝐻 (2.163) using the Baker-Campbell-Hausdorff (BCH) identity
𝑒 −𝑇 𝐻𝑒𝑇 = 𝑒 ad𝑇 𝐻
∞
Õ 1 𝑛
= ad 𝐻
𝑛=0
𝑛! 𝑇
1
= 𝐻 + [𝐻, 𝑇] + [[𝐻, 𝑇], 𝑇] + · · · , (2.167)
2
where ad𝑎 𝑏 = [𝑏, 𝑎]. When one plugs this expansion into energy expression (2.161) it
can be shown that, when the reference state |Φ0 i is a single determinant, the expansion
terminates at fourth order (𝑛 = 4). Coupled cluster with single determinant reference states
usually performs well at equilibrium configurations but poorly for strongly correlated
systems. This shortcoming can be overcome with unitary coupled cluster theory.
2.4.6 Pair Coupled Cluster Doubles Theory
In pair coupled cluster doubles theory [42], often abbreviated as pCCD, the cluster
operator (2.158) is set to a variant of 𝑇2 called 𝑇𝑝 which is restricted to moving pairs of
fermions. That is, 𝑇 = 𝑇𝑝 , where
Õ
𝑇𝑝 = 𝑡𝑖𝑎 𝐴†𝑎 𝐴𝑖 . (2.168)
𝑖𝑎
30
Here, 𝐴† and 𝐴 are the pair fermionic creation and annihilation operators, respectively,
defined as
𝐴†𝑝 = 𝑎 †𝑝+ 𝑎 †𝑝− (2.169)
𝐴 𝑝 = 𝑎 𝑝− 𝑎 𝑝+ , (2.170)
where 𝑝+ and 𝑝− index the spin-up and spin-down fermions in the 𝑝 th energy level,
respectively. The coupled cluster equations (2.161 and 2.162) become
𝐸 = hΦ0 |𝐻|Φ0 i (2.171)
0 = Φ𝑖𝑎 𝐻 Φ0 , (2.172)
where the similarity transformed Hamiltonian 𝐻 is
𝐻 = 𝑒 −𝑇𝑝 𝐻𝑒𝑇𝑝 , (2.173)
and the excited state hΦ𝑖𝑎 | is obtained by
hΦ𝑖𝑎 | = hΦ0 | 𝐴𝑖† 𝐴𝑎 . (2.174)
Note that we truncated the amplitude equation (2.162) to a single excited state for the pair
coupled cluster doubles amplitude equations.
2.4.7 Unitary Coupled Cluster Theory
In unitary coupled cluster (UCC) theory [8], the cluster operator 𝑇 is replaced with the
purely imaginary operator 𝑇 − 𝑇 † , leading to the unitary exponential ansatz
†
|Ψi = 𝑒𝑇−𝑇 |Φ0 i . (2.175)
31
This ansatz is variational and thus the ground state energy 𝐸 0 can be obtained from the
variational principle
𝐸 0 = min hΦ0 |𝐻|Φ0 i , (2.176)
𝑡
where 𝑡 are the cluster amplitudes, of which 𝑇 is a function, and 𝐻 is the similarity
transformed Hamiltonian
† −𝑇 †
𝐻 = 𝑒𝑇 𝐻𝑒𝑇−𝑇 . (2.177)
While unitary coupled cluster theory overcomes some weaknesses of coupled cluster theory,
it is not classically tractable to implement. One can see this by expanding, analogously to
(2.167), the similarity transformed Hamiltonian for UCC using the BCH identity,
† −𝑇 †
𝑒𝑇 𝐻𝑒𝑇−𝑇 =𝑒 ad𝑇 −𝑇 † 𝐻 (2.178)
∞
Õ 1 𝑛
= ad𝑇−𝑇 † 𝐻
𝑛=0
𝑛!
1
= 𝐻 + [𝐻, 𝑇] − [𝐻, 𝑇 † ] + ( [[𝐻, 𝑇], 𝑇] + [[𝐻, 𝑇 † ] , 𝑇 † ]
2
− [[𝐻, 𝑇] , 𝑇 † ] − [[𝐻, 𝑇 † ] , 𝑇] ) + · · · , (2.179)
which, unlike the expansion for CC, does not naturally truncate. Thus, it is classically
intractable to solve the resulting amplitude and energy equations. Fortunately, unitary
operators are implementable on quantum computers (indeed, as all quantum gates must be
unitary). Thus, UCC is a viable candidate ansatz for the so-called Variational Quantum
Eigensolver (VQE), a quantum algorithm which is discussed in the following chapter.
32
CHAPTER 3
QUANTUM COMPUTING
3.1 Introduction
The idea of quantum computing was first put forth in 1982 by Richard Feynman [33].
The initial motivation was to simulate quantum mechanics itself because, as Feynman said
“Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better
make it quantum mechanical...” Quantum computers use qubits, rather than bits, as their
fundamental units. A qubit is an abstraction of a quantum binary event. The states qubits
are manipulated by physical processes whose abstractions are referred to as quantum gates.
Today’s quantum computers have of order 100 qubits and can implement of order 1000
gates [19] before the accrued noise becomes insurmountable.
3.2 Qubits
The bit is the basic unit of information in classical computing. It is the abstraction
of a binary logical state. One denotes the two possible states as 0 and 1. That is, a bit 𝑐 is
𝑐 = 0 OR 1. (3.1)
A bit can be realized physically by a classical binary event, a classical event that has exactly
two distinct outcomes. Examples include the toss of a coin, the presence or absence of a
hole in a paper card, or (as used in modern computers) two levels of electric charge stored
in a capacitor. Multiple bits are written together as a bit-string
𝑐 = 𝑐 0 𝑐 2 . . . 𝑐 𝑛−1 , (3.2)
where 𝑐𝑖 = 0, 1 for 𝑖 = 0, 1, . . . , 𝑛 − 1. Thus, 𝑛 qubits can represent 2𝑛 distinct logical
states.
33
The qubit, by contrast, is the basic unit of information in quantum computing. It is
the abstraction of a quantum binary event. Examples include an electron being spin-up or
spin-down or a quantum system being in one of two energy states. The state of a qubit is
spanned by the computational basis which consists of the following two quantum states
©1 ª
|0i = ®® , (3.3)
0
« ¬
©0 ª
|1i = ®® . (3.4)
1
« ¬
Thus, a qubit is a two-level quantum state which can be written as a superposition of the
computational basis states, |0i and |1i. That is, a qubit |𝑞i can be written as
|𝑞i = 𝑎 |0i + 𝑏 |1i , (3.5)
where 𝑎, 𝑏 ∈ C, under the restriction that
|𝑎| 2 + |𝑏| 2 = 1. (3.6)
One convenient way to represent the state of a single qubit |𝑞i is on the Bloch sphere [4].
To do so, one parameterizes the state’s coefficients with two angles 𝜃 and 𝜙 as
𝜃 𝜃
|𝜓i = cos |0i + sin 𝑒𝑖𝜙 |1i . (3.7)
2 2
Then, the qubit |𝜓i can be represented on the Bloch sphere as a vector with polar angle 𝜃
and azimuthal angle 𝜙 as depicted in Figure 3.1. Note that the coefficient of the |0i term in
3.7 does note contain a complex phase. This is because the global phase of a qubit’s state
is irrelevant and thus the relative phase of the |0i state can always be absorbed into such a
global phase.
34
Figure 3.1: Depiction of the state |𝜓i on the Bloch sphere [20].
As two by one, complex-valued vectors, qubits can be “multiplied" in several important
ways. To explain these manipulations, we must first introduce the adjoint of a qubit. Given
a qubit
©𝑎 ª
|𝑞i = ®® , (3.8)
𝑏
« ¬
its adjoint h𝑞| is given by its Hermitian conjugate
†
h𝑞| = |𝑞i = 𝑎∗ 𝑏∗ . (3.9)
The inner-product of two qubits |𝑞 1 i and |𝑞 2 i is defined as
©𝑎 2 ª
h𝑞 1 |𝑞 2 i = 𝑎 ∗ 𝑏 ∗ ®® = 𝑎 ∗1 𝑎 2 + 𝑏 ∗1 𝑏 2 . (3.10)
1 1
𝑏
« 2¬
35
The outer-product of two qubits |𝑞 1 i and |𝑞 2 i is defined as
©𝑎 1 ª ©𝑎 1 𝑎 ∗ 𝑎 1 𝑏 ∗ ª
∗ ∗ 2 2®
|𝑞 1 i h𝑞 2 | = ® 𝑎 𝑏 =
®
®. (3.11)
2 2
𝑏1 𝑏 1 𝑎 ∗2 𝑏 1 𝑏 ∗2
« ¬ « ¬
Finally, the tensor-product of two-qubits is defined as
© © 𝑎 2 ªª © 𝑎 1 𝑎 2 ª
𝑎 1 ®® ®
®® ®
©𝑎 1 ª ©𝑎 2 ª 𝑏 2 ® 𝑎 1 𝑏 2 ®
®
|𝑞 1 𝑞 2 i = |𝑞 1 i ⊗ |𝑞 2 i = ®® ⊗ ®® = « ¬® = ®.
® (3.12)
®
𝑏1 𝑏2 ©𝑎 2 ª® 𝑏 1 𝑎 2 ®®
« ¬ « ¬ 𝑏 1 ®® ®
®®
𝑏2 𝑏 1 𝑏 2
« « ¬¬ « ¬
3.3 Quantum Gates
Quantum gates are abstractions of the physical actions applied to the physical systems
representing qubits. Mathematically, they are complex-valued, unitary matrices which act
on the complex-valued, normalized vectors that represent qubits. As the quantum analog
of classical logic gates (such as AND and OR), there is a corresponding quantum gate for
every classical gate; however, there are quantum gates which have no classical counterpart.
They act on a set of qubits, changing the qubit’s state in the process. That is, if 𝑈 is a
quantum gate and |𝑞i is a qubit, then acting the gate 𝑈 on the qubit |𝑞i transforms the qubit
as follows:
𝑈
|𝑞i → 𝑈 |𝑞i . (3.13)
This action can be represented via the following quantum circuit
|𝑞i 𝑈 𝑈 |𝑞i (3.14)
36
Quantum circuits are diagrammatic representations of quantum algorithms. The horizontal
dimension corresponds to time; moving left to right corresponds to forward motion in time.
They consist of a set of qubits |𝑞 𝑛 i which are stacked vertically on the left-hand side of
the diagram. Lines, called quantum wires, extend horizontally to the right from each
qubit, representing its state moving forward in time. Additionally, they contain a set of
quantum gates that interrupt to the quantum wires, implying that they are applied to the
corresponding qubit. Gates are applied chronologically, left to right. With this, we can see
that the quantum circuit above (3.15) implies that the quantum gate 𝑈 is being applied to a
qubit in state |𝑞i.
To explain what quantum circuits represent mathematically, consider the following
circuit
|𝑞 0 i 𝐴 𝐵 (3.15)
|𝑞 1 i 𝐶 𝐷
which implies the following mathematical statement
|𝑞 0 𝑞 1 i → (𝐵 ⊗ 𝐷)( 𝐴 ⊗ 𝐶) |𝑞 0 𝑞 1 i (3.16)
→ (𝐵𝐴) ⊗ (𝐷𝐶) |𝑞 0 𝑞 1 i (3.17)
→ 𝐵𝐴 |𝑞 0 i 𝐷𝐶 |𝑞 1 i . (3.18)
Note that mathematical form is in reverse order from circuit form (𝐵𝐴 ↔ 𝐴𝐵). This is
because the operator closest to the state mathematically (furthest to the right on a quantum
circuit) acts first. Additionally, we are able to write the actions of the top two and bottom
two gates as acting separately on each qubit as every gate here is a single-qubit gate (acting
on only one qubit). The same would not be true for certain two-qubit gates which entangle
37
the states of the two qubits, not allowing their state to be written in a separable form.
Finally, we define the depth of a quantum circuit as the number of columns of gates. The
circuit above thus has a depth of two because it contains two columns of gates, namely
𝐴 ⊗ 𝐶 and 𝐵 ⊗ 𝐷.
3.3.1 Single-Qubit Gates
A single-qubit gate is the abstraction of a physical action that is applied to one qubit.
It can be represented by a matrix 𝑈 ∈ SU(2). Any single-qubit gate can be parameterized
by three angles: 𝜃, 𝜙, and 𝜆 as follows
© cos 2𝜃 −𝑒𝑖𝜆 sin 2𝜃 ª
𝑈 (𝜃, 𝜙, 𝜆) = ®.
® (3.19)
𝑖𝜙 𝜃 𝑖(𝜙+𝜆) 𝜃
𝑒 sin 2 𝑒 cos 2
« ¬
A common set of single-qubit gates is the set of Pauli gates, whose members correspond
to the Pauli matrices
©1 0 ª
𝐼 = ®,
® (3.20)
0 1
« ¬
©0 1 ª
𝑋 = ®,
® (3.21)
1 0
« ¬
©0 −𝑖 ª
𝑌 = ®,
® (3.22)
𝑖 0
« ¬
©1 0 ª
𝑍 = ®,
® (3.23)
0 −1
« ¬
which satisfy the relation
[𝜎, 𝜏] = 𝑖𝜖 𝜎𝜏𝜐 𝜐, (3.24)
38
for 𝜎, 𝜏, 𝜐 ∈ {𝑋, 𝑌 , 𝑍 }. These gates form a basis for the algebra 𝔰𝔲(2). Exponentiating
them will thus give us a basis for SU(2), the group within which all single-qubit gates live.
These exponentiated Pauli gates are called rotation gates 𝑅𝜎 (𝜃) because they rotate the
quantum state around the axis 𝜎 = 𝑋, 𝑌 , 𝑍 of the Bloch sphere (Figure 3.1) by an angle 𝜃.
They are defined as
−𝑖 2𝜃 𝑋
© cos 2𝜃 −𝑖 sin 2𝜃 ª
𝑅 𝑋 (𝜃) = 𝑒 = ®,
® (3.25)
𝜃 𝜃
−𝑖 sin 2 cos 2
« ¬
−𝑖 2𝜃 𝑌
©cos 2𝜃 − sin 2𝜃 ª
𝑅𝑌 (𝜃) = 𝑒 = ®,
® (3.26)
𝜃 𝜃
sin 2 cos 2
« ¬
−𝑖 2𝜃 𝑍
©𝑒 −𝑖𝜃/2 0 ª
𝑅 𝑍 (𝜃) = 𝑒 = ®.
® (3.27)
0 𝑒 𝑖𝜃/2
« ¬
Because they form a basis for SU(2), any single-qubit gate can be decomposed into three
rotation gates. Indeed
©𝑒 −𝑖𝜙/2 0 ª ©cos 2𝜃 − sin 2𝜃 ª ©𝑒 −𝑖𝜆/2 0 ª
𝑅𝑧 (𝜙)𝑅 𝑦 (𝜃)𝑅𝑧 (𝜆) = ®
®
®
®
®
® (3.28)
𝑖𝜙/2 𝜃 𝜃 𝑖𝜆/2
0 𝑒 sin 2 cos 2 0 𝑒
« ¬« ¬« ¬
−𝑖(𝜙+𝜆)/2
© cos 2𝜃 −𝑒𝑖𝜆 sin 2𝜃 ª
=𝑒 𝑖𝜙
®,
® (3.29)
𝜃 𝑖(𝜙+𝜆) 𝜃
𝑒 sin 2 𝑒 cos 2
« ¬
which is, up to a global phase, equal to the expression for an arbitrary single-qubit gate
(3.19).
3.3.2 Two-Qubit Gates
A two-qubit gate is the abstraction of a physical action that is applied to two qubits. It
can be represented by a matrix 𝑈 from the group SU(4). One important class of two-qubit
39
gates are the controlled gates, which work as follows: Suppose 𝑈 is a single-qubit gate. A
controlled-𝑈 gate (𝐶𝑈) acts on two qubits: a control qubit |𝑥i and a target qubit |𝑦i. The
controlled-𝑈 gate applies the identity 𝐼 or the single-qubit gate 𝑈 to the target qubit if the
control gate is in the zero state |0i or the one state |1i, respectively. The control qubit is
acted upon by the identity 𝐼. This can be represented as follows:
|𝑥𝑦i if |𝑥i = |0i
𝐶𝑈 |𝑥𝑦i = . (3.30)
|𝑥i 𝑈 |𝑦i if |𝑥i = |1i
The action of a controlled-𝑈 gate (𝐶𝑈) can be represented via quantum circuit as follows
|𝑥i • |𝑥i (3.31)
|𝑦i , |𝑥i = |0i
|𝑦i 𝑈
𝑈 |𝑦i , |𝑥i = |1i
It can be written in matrix form by writing it as a superposition of the two possible cases,
each written as a simple tensor product
𝐶𝑈 = |0i h0| ⊗ 𝐼 + |1i h1| ⊗ 𝑈 (3.32)
©1 0 0 0 ª
®
®
0 1 0 0 ®
= ®.
® (3.33)
0 0 𝑢 00 𝑢 01 ®®
®
«0 0 𝑢 10 𝑢 11 ¬
where 𝑢𝑖 𝑗 (for 𝑖, 𝑗 ∈ 0, 1) are the matrix elements of 𝑈. One of the most fundamental
controlled gates is the CNOT gate. It is defined as the controlled-𝑋 gate 𝐶 𝑋 and thus flips
40
the state of the target qubit if the control qubit is in the zero state |0i. It can be written in
matrix form as follows:
©1 0 0 0ª
®
®
0 1 0 0®
CNOT = ®.
® (3.34)
0 0 0 1®®
®
«0 0 1 0¬
A widely used two-qubit gate that goes beyond the simple controlled function is the SWAP
gate. It swaps the states of the two qubits that it acts upon
SWAP |𝑥𝑦i = |𝑦𝑥i , (3.35)
as depicted in the quantum circuit below
|𝑥i × |𝑦i (3.36)
|𝑦i × |𝑥i
and has the following matrix form
©1 0 0 0ª
®
®
0 0 1 0®
SWAP = ®.
® (3.37)
0 1 0 0®®
®
«0 0 0 1¬
It can be decomposed into a series of three CNOTs, each of which has its directionality
flipped from the previous
|𝑥i • • |𝑦i (3.38)
|𝑦i • |𝑥i
41
As for arbitrary two-qubit gates 𝑈 ∈ SU(4), they can be optimally decomposed (up to a
global phase) into the following sequence [83] fifteen elementary one-qubit gates and three
CNOT gates
𝑈1 • 𝑅 𝑦 (𝜃 1 ) 𝑅 𝑦 (𝜃 2 ) • 𝑈3 (3.39)
𝑈2 𝑅𝑧 (𝜃 3 ) • 𝑈4
where 𝑈1 , 𝑈2 , 𝑈3 , 𝑈4 are single-qubit gates, each of which requires three parameters as each
can be decomposed into three elementary one-qubit gates (rotation gates). Additionally,
the parameters 𝜃 1 , 𝜃 2 , 𝜃 3 are determined by the arbitrary two-qubit gate to be decomposed.
Two-qubit gates that are restricted to 𝑈 ∈ SO(4) can be decomposed into a shorter depth
circuit consisting of just twelve elementary single-qubit gates and two CNOT gates
𝑅𝑧 (𝜋/2) 𝑅 𝑦 (𝜋/2) • 𝑈1 • 𝑅 ∗𝑦 (𝜋/2) 𝑅𝑧∗ (𝜋/2)
𝑅𝑧 (𝜋/2) 𝑈2 𝑅𝑧∗ (𝜋/2)
(3.40)
3.4 Variational Quantum Eigensolver
3.4.1 Introduction
One of the first algorithms developed to estimate the eigenenergies of a Hamiltonian
was quantum phase estimation [50]. In the algorithm, one encodes the eigenenergies, one
binary bit at a time (up to 𝑛 bits), into the complex phases of the quantum states of the Hilbert
space for 𝑛 qubits. It does this by applying powers of controlled unitary evolution operators
to a quantum state that can be expanded in terms of the Hamiltonian’s eigenvectors. The
42
eigenenergies are encoded into the complex phases in such a way that taking the inverse
quantum Fourier transformation of the states into which the eigenenergies are encoded
results in a measurement probability distribution that has peaks around the bit strings that
represent a binary fraction which corresponds to the eigenenergies of the quantum state
acted upon by the controlled unitary operators. While quantum phase estimation (QPE) is
provably efficient, non-hybrid, and non-variational, the number of qubits and circuit length
required to execute it is too great for our NISQ era quantum computers. Thus, QPE will
only be efficiently applicable on large, fault-tolerant quantum computers that likely won’t
exist in the near future.
Therefore, a different algorithm for finding the eigenenergies of a quantum Hamiltonian
was put forth in 2014 called the variational quantum eigensolver [63], commonly referred
to as VQE. The algorithm is hybrid, meaning that it requires the use of both a quantum
computer and a classical computer. It is also variational, meaning that it relies, ultimately,
on solving an optimization problem by varying parameters and thus is not deterministic
like QPE. The variational quantum eigensolver is based on the variational principle: The
expectation value of a Hamiltonian 𝐻 in a state |𝜓(𝜃)i parameterized by a set of angles 𝜃,
is always greater than or equal to the minimum eigenenergy 𝐸 0 . To see this, let |𝑛i be the
eigenstates of 𝐻
𝐻|𝑛i = 𝐸 𝑛 |𝑛i. (3.41)
We can then expand our state |𝜓(𝜃)i in terms of said eigenstates
Õ
|𝜓(𝜃)i = 𝑐 𝑛 |𝑛i, (3.42)
𝑛
43
and take the expectation value of the Hamiltonian in this state to yield
Õ
h𝜓(𝜃)|𝐻|𝜓(𝜃)i = 𝑐∗𝑚 𝑐 𝑛 h𝑚|𝐻|𝑛i
𝑛𝑚
Õ
= 𝑐∗𝑚 𝑐 𝑛 𝐸 𝑛 h𝑚|𝑛i
𝑛𝑚
Õ
= 𝛿𝑛𝑚 𝑐∗𝑚 𝑐 𝑛 𝐸 𝑛
𝑛𝑚
Õ
= |𝑐 𝑛 | 2 𝐸 𝑛
𝑛
Õ
≥ 𝐸0 |𝑐 𝑛 | 2
𝑛
= 𝐸0, (3.43)
which implies that we can minimize over the set of angles 𝜃 and arrive at the ground state
energy 𝐸 0 :
min h𝜓(𝜃)|𝐻|𝜓(𝜃)i = 𝐸 0 . (3.44)
𝜃
Figure 3.2: Schematic of the Variational Quantum Eigensolver.
Using this fact, VQE can be broken down into the following steps, as noted in Figure
3.2:
44
1. Prepare the variational state |𝜓(𝜃)i on a quantum computer.
2. Measure this circuit in various bases and send these measurements to a classical
computer.
3. Post-processes the measurement data on the classical computer to compute the
expectation value h𝜓(𝜃)|𝐻|𝜓(𝜃)i
4. Vary the parameters 𝜃 according to a classical minimization algorithm and send them
back to the quantum computer which runs step 1 again.
This loop continues until the classical optimization algorithm terminates which results in
a set of angles 𝜃 min that characterize the ground state |𝜙(𝜃 min )i and an estimate for the
ground state energy 𝐸 0 = h𝜓(𝜃 min )|𝐻|𝜓(𝜃 min )i.
3.4.2 Expectation Values
To execute the second step of VQE, we need to understand how expectation values of
operators can be estimated via quantum computers by post-processing measurements of
quantum circuits in different basis. To rotate bases, one uses the basis rotator 𝐵𝜎 which is
defined for each Pauli gate 𝜎 to be
if 𝜎 = 𝑋
𝐻,
𝐵𝜎 = 𝐻𝑆 † , if 𝜎 = 𝑌 . (3.45)
𝐼,
if 𝜎 = 𝑍
Note the following identity of the basis rotator
𝐵†𝜎 𝑍 𝐵𝜎 = 𝜎, (3.46)
45
which follows from the fact that 𝐻𝑍 𝐻 = 𝑋 and 𝑆𝑋𝑆 † = 𝑌 . With this, we see that the
expectation value of an arbitrary Puali-gate 𝜎 in the state |𝜓i can be expressed as a linear
combination of probabilities
𝐸 𝜓 (𝜎) = h𝜓|𝜎|𝜓i
= h𝜓|𝐵†𝜎 𝑍 𝐵𝜎 |𝜓i
= h𝜙|𝑍 |𝜙i
© Õ
= h𝜙| (−1) 𝑥 |𝑥i h𝑥| ® |𝜙i
ª
«𝑥∈{0,1} ¬
Õ
2
= (−1) 𝑥 |h𝑥|𝜙i|
𝑥∈{0,1}
Õ
= (−1) 𝑥 𝑃(|𝜙i → |𝑥i), (3.47)
𝑥∈{0,1}
where |𝜙i = |𝐵𝜎 𝜙i and 𝑃(|𝜙i → |𝑥i) is the probability that the state |𝜙i collapses to the
state |𝑥i when measured. This can be extended to any arbitrary Pauli string as follows:
Ë
consider the string of Pauli operators 𝑃 = 𝑝∈𝑄 𝜎𝑝 which acts non-trivially on the set of
qubits 𝑄 which is a subset of the total set of 𝑛 qubits in the system. Then
©Ì ª
𝐸 𝜓 (𝑃) = h𝜓| 𝜎𝑝 ® |𝜓i
« 𝑝∈𝑄 ¬
©Ì ª ©Ì ª
= h𝜓| 𝜎𝑝 ® 𝐼𝑞 ® |𝜓i
« 𝑝∈𝑄 ¬ « 𝑞∉𝑄 ¬
©Ì † ª ©Ì ª
= h𝜓| 𝐵 𝜎𝑝 𝑍 𝑝 𝐵 𝜎𝑝 ® 𝐼𝑞 ® |𝜓i
« 𝑝∈𝑄 ¬« 𝑞∉𝑄 ¬
©Ì † ª ©Ì ª ©Ì ª ©Ì
= h𝜓| 𝐵𝜎𝑝 ® |𝜓i
ª
𝐵 𝜎𝑝 ® 𝑍𝑝® 𝐼𝑞 ®
« 𝑝∈𝑄 ¬ « 𝑝∈𝑄 ¬ « 𝑞∉𝑄 ¬ « 𝑝∈𝑄 ¬
46
©Ì ª ©Ì ª
= h𝜙| 𝑍𝑝® 𝐼𝑞 ® |𝜙i
« 𝑝∈𝑄 ¬« 𝑞∉𝑄 ¬
©Ì Õ ª ©Ì Õ
= h𝜙| (−1) 𝑥 𝑝 𝑥 𝑝 𝑥 𝑝 ® 𝑦 𝑞 ® |𝜙i
ª
𝑦𝑞
« 𝑝∈𝑄 𝑥 𝑝 ∈{0 𝑝 ,1 𝑝 } ¬ « 𝑞∉𝑄 𝑦 𝑞 ∈{0𝑞 ,1𝑞 } ¬
© Õ Í
= h𝜙| (−1) 𝑝 ∈𝑄 𝑥 𝑝 |𝑥i h𝑥| ® |𝜙i
ª
«𝑥∈{0,1}𝑛 ¬
Õ Í
= (−1) 𝑝 ∈𝑄 𝑥 𝑝 |h𝑥|𝜙i| 2
𝑥∈{0,1} 𝑛
Õ Í
𝑥𝑝
= (−1) 𝑝 ∈𝑄 𝑃(|𝜙i → |𝑥i), (3.48)
𝑥∈{0,1} 𝑛
Ë E
where |𝜙i = 𝑝∈𝑄 𝐵 𝜎 𝑝 𝜓 . Finally, because the expectation value is linear
!
Õ Õ
𝐸𝜓 𝜆 𝑚 𝑃𝑚 = 𝜆 𝑚 𝐸 𝜓 (𝑃𝑚 ), (3.49)
𝑚 𝑚
one can estimate any observable that can be written as a linear combination of Puali-string
terms.
3.4.3 Measurement
To estimate the probability 𝑃(|𝜙i → |𝑥i) from the previous section, one prepares the
state |𝜙i on a quantum computer and measures it, and then repeats this process (prepare
and measure) several times. The probability 𝑃(|𝜙i → |𝑥i) is estimated to be the number
of times that one measures the bit-string 𝑥 divided by the total number of measurements
that one makes; that is
𝑀
Õ 𝑥𝑚
𝑃(|𝜙i → |𝑥i) ≈ , (3.50)
𝑚=1
𝑀
47
where
1 if the result of measurement is 𝑥
𝑥𝑚 = (3.51)
0 if the result of measurement is not 𝑥.
By the law of large numbers [11], the approximation (3.52) approaches equality as 𝑀 goes
to infinity
𝑀
Õ 𝑥𝑚
𝑃(|𝜙i → |𝑥i) = lim . (3.52)
𝑀→∞
𝑚=1
𝑀
As we obviously do not have infinite time nor infinite quantum computers to be run in
parallel, we must truncate our number of measurement 𝑀 to a finite, but sufficiently large
number. More precisely, for precision 𝜖, each expectation estimation subroutine within
VQE requires O (1/𝜖 2 ) samples from circuits of depth O (1) [85].
3.5 Transformations
While many-body nuclear physics operators are written in terms of fermionic operators,
quantum computers work with Pauli operators. Thus, in order to simulate many-body
nuclear physics on a quantum computer, we need a transformation between the two sets
of operators. Several such transformations exist [74] (each with their own advantages and
disadvantages) of which we list three here: Jordan-Wigner, Parity-Basis, and Bravyi-Kitiav.
Transformation Basis # of Operators Locality
Jordan-Wigner Occupation Number O (𝑁) Local
Parity-Basis Parity O (𝑁) Local
Bravyi-Kitiav Mixed O (log(𝑁)) Non-local
Table 3.1: Comparison of basis, number of operators, and locality the of Jordan-Wigner,
parity-basis, and Bravyi-Kitiav transformations.
Jordan-Wigner works in the occupation number representation which is naturally map-
able to the computational basis set of a quantum computer. However, it requires long
48
strings of operators. Its main advantage for near-term devices is that it is local and hence
implementable on quantum computers with linear qubit-connectivity. The parity-basis
transformation is similar to Jordan-Wigner except that it works in the parity-basis. The
Bravyi-Kitiav transformation is a mix between the Jordan-Wigner and the parity-basis
transformations that allows for a shorter number of operators but which comes with the
cost of being non-local and hence not suitable for quantum computers with limited qubit-
connectivity. In this work, we use the Jordan-Wigner transformation because, although
it requires more operators (O (𝑁)) it uses the occupation number representation, which is
naturally implementable on a quantum computer. Additionally, it is local, making it more
easily implementable on devices with limited qubit connectivity, which describes most
near-term devices. Here, qubit connectivity refers to which qubits are connected to which
other qubits. Two qubits are connected if one can implement a two-qubit gate between
them.
3.5.1 Jordan-Wigner Transformation
The Jordan-Wigner transformation was originally developed by Pascual Jordan and
Eugene Wigner for one-dimensional lattice models [48]. The transformation is a mapping
between fermionic and Pauli operators which stores information locally in the occupation
number basis. It is given below as
𝑝−1
!
Ö
𝑎 †𝑝 = 𝑍𝑛 𝑄 −𝑝 , (3.53)
𝑛=1
𝑝−1
!
Ö
𝑎𝑝 = 𝑍𝑛 𝑄 +𝑝 , (3.54)
𝑛=1
where
𝑋 𝑝 ± 𝑖𝑌𝑝
𝑄 ±𝑝 = , (3.55)
2
49
and
𝑝−1
! 𝑁
Ì ©Ì ª
𝜎𝑝 = 𝐼 ⊗𝜎⊗ 𝐼® , (3.56)
𝑛=1 «𝑛=𝑝+1 ¬
where 𝜎 = 𝐼, 𝑋, 𝑌 , 𝑍 is a Pauli operator and 𝑁 is the number of qubits in the system. To
gain some intuition for the mapping, note that the action on many-fermionic states (2.37)
is preserved by the transformation
𝑝−1
!
Ö
𝑎𝑖† |𝑛1 . . . 𝑛𝑛 i = 𝑍𝑛 𝑄 −𝑝 |𝑛1 . . . 𝑛𝑛 i = (−1) 𝑁 𝑝 (1 − 𝑛 𝑝 ) 𝑛1 . . . 𝑛 𝑝−1 1𝑛 𝑝+1 . . . 𝑛𝑛 ,
𝑛=1
(3.57)
𝑝−1
!
Ö
𝑎𝑖 |𝑛1 . . . 𝑛𝑛 i = 𝑍𝑛 𝑄 +𝑝 |𝑛1 . . . 𝑛𝑛 i = (−1) 𝑁 𝑝 𝑛 𝑝 𝑛1 . . . 𝑛 𝑝−1 0𝑛 𝑝+1 . . . 𝑛𝑛 , (3.58)
𝑛=1
Í 𝑝−1
where 𝑁 𝑝 = 𝑚=1 𝑛𝑚 , since
𝑍 |𝑛i = (−1) 𝑛 |𝑛i , (3.59)
𝑄 − |𝑛i = (1 − 𝑛) |1i , (3.60)
𝑄 + |𝑛i = 𝑛 |0i , (3.61)
where 𝑛 = 0, 1. The mapping holds because it obeys the fermionic anti-commutation
relations (2.39) as verified below: First, consider the case 𝑝 = 𝑞. In this case, the anti-
commutation relations are
( 𝑝−1 ! 𝑝−1
! ) 𝑝−1
!
Ö Ö Ö
{𝑎 𝑝 , 𝑎 †𝑝 } = 𝑍𝑛 𝑄 +𝑝 , 𝑍𝑛 𝑄 −𝑝 = {𝑍𝑛 , 𝑍𝑛 } {𝑄 +𝑝 , 𝑄 −𝑝 } = 𝐼 𝑝 (3.62)
𝑛=1 𝑛=1 𝑛=1
( 𝑝−1
! 𝑝−1
! ) 𝑝−1
!
Ö Ö Ö
{𝑎 𝑝 , 𝑎 𝑝 } = 𝑍𝑛 𝑄 +𝑝 , 𝑍𝑛 𝑄 +𝑝 = {𝑍𝑛 , 𝑍𝑛 } {𝑄 +𝑝 , 𝑄 +𝑝 } = 0 (3.63)
𝑛=1 𝑛=1 𝑛=1
( 𝑝−1
! 𝑝−1
! ) 𝑝−1
!
Ö Ö Ö
{𝑎 †𝑝 , 𝑎 †𝑝 } = 𝑍𝑛 𝑄 −𝑝 , 𝑍𝑛 𝑄 −𝑝 = {𝑍𝑛 , 𝑍𝑛 } {𝑄 −𝑝 , 𝑄 −𝑝 } = 0, (3.64)
𝑛=1 𝑛=1 𝑛=1
50
which follow from the fact that
1
{𝑄 ±𝑝 , 𝑄 ∓𝑝 } = {𝑋 𝑝 ± 𝑖𝑌𝑝 , 𝑋 𝑝 ∓ 𝑖𝑌𝑝 }
4
1
= ({𝑋 𝑝 , 𝑋 𝑝 } ∓ 𝑖{𝑋 𝑝 , 𝑌𝑝 } ± 𝑖{𝑌𝑝 , 𝑋 𝑝 } + {𝑌𝑝 , 𝑌𝑝 } ) = 𝐼 𝑝 , (3.65)
4
while
1
{𝑄 ±𝑝 , 𝑄 ±𝑝 } = {𝑋 𝑝 ± 𝑖𝑌𝑝 , 𝑋 𝑝 ± 𝑖𝑌𝑝 }
4
1
= ({𝑋 𝑝 , 𝑋 𝑝 } ± 𝑖{𝑋 𝑝 , 𝑌𝑝 } ± 𝑖{𝑌𝑝 , 𝑋 𝑝 } − {𝑌𝑝 , 𝑌𝑝 } ) = 0. (3.66)
4
Second, consider the case 𝑝 ≠ 𝑞. Without loss of generality, we can set 𝑝 < 𝑞. In this
case, the anti-commutation relations are
( 𝑝−1 ! 𝑞−1
! )
Ö Ö
{𝑎 𝑝 , 𝑎 †𝑞 } = 𝑍𝑛 𝑄 +𝑝 , 𝑍𝑛 𝑄 −𝑝 (3.67)
𝑛=1 𝑛=1
𝑝−1
! 𝑞−1
Ö © Ö
= {𝑍𝑛 , 𝑍𝑛 } {𝑄 +𝑝 , 𝑍 𝑝− } {𝐼𝑛 , 𝑍𝑚 }® {𝐼𝑞 , 𝑄 −𝑞 } = 0
ª
(3.68)
𝑛=1 «𝑚=𝑝+1 ¬
( 𝑝−1
! 𝑞−1
! )
Ö Ö
{𝑎 𝑝 , 𝑎 𝑞 } = 𝑍𝑛 𝑄 +𝑝 , 𝑍𝑛 𝑄 +𝑝 (3.69)
𝑛=1 𝑛=1
𝑝−1
! 𝑞−1
Ö © Ö
= {𝑍𝑛 , 𝑍𝑛 } {𝑄 +𝑝 , 𝑍 𝑝− } {𝐼𝑛 , 𝑍𝑚 }® {𝐼𝑞 , 𝑄 +𝑞 } = 0
ª
(3.70)
𝑛=1 «𝑚=𝑝+1 ¬
( 𝑝−1
! 𝑞−1
! )
Ö Ö
{𝑎 †𝑝 , 𝑎 †𝑞 } = 𝑍𝑛 𝑄 −𝑝 , 𝑍𝑛 𝑄 −𝑝 (3.71)
𝑛=1 𝑛=1
𝑝−1
! 𝑞−1
Ö © Ö
= {𝑍𝑛 , 𝑍𝑛 } {𝑄 −𝑝 , 𝑍 𝑝− } {𝐼𝑛 , 𝑍𝑚 }® {𝐼𝑞 , 𝑄 −𝑞 } = 0,
ª
(3.72)
𝑛=1 «𝑚=𝑝+1 ¬
which follow from the fact that
1 1
𝑄 ±𝑝 , 𝑍 𝑝 =
{𝑋 ∓ 𝑖𝑌 , 𝑍 } = ({𝑋, 𝑍 } ∓ 𝑖{𝑌 , 𝑍 }) = 0 (3.73)
2 2
51
3.5.2 Pair Jordan-Wigner Transformation
The Jordan-Wigner transformation is simplified when dealing with pair fermionic
operators:
𝐴†𝑝 = 𝑄 −𝑝 , (3.74)
𝐴 𝑝 = 𝑄 +𝑝 , (3.75)
𝑁𝑝 = 𝐼𝑝 − 𝑍𝑝. (3.76)
Namely, the string of 𝑍 operators preceding the 𝑄 ± operator is dropped. This is one of the
main advantages of working with the nuclear pairing model. Because it can be written in
terms of pair fermionic operators, its mapping to quantum operators is greatly simplified.
This mapping holds because it obeys the pair fermionic commutation relations (5.6 - 5.8)
as verified below:
[ 𝐴 𝑝 , 𝐴†𝑞 ] = [𝑄 +𝑝 , 𝑄 −𝑞 ]
1
= [𝑋 𝑝 + 𝑖𝑌𝑝 , 𝑋𝑞 − 𝑖𝑌𝑞 ]
4
1
= ([𝑋 𝑝 , 𝑋𝑞 ] − 𝑖[𝑋 𝑝 , 𝑌𝑞 ] + 𝑖[𝑌𝑝 , 𝑋𝑞 ] + [𝑌𝑝 , 𝑌𝑞 ] )
4
= 𝛿 𝑝𝑞 𝑍 𝑝
= 𝛿 𝑝𝑞 (𝐼 𝑝 − 𝑁 𝑝 ), (3.77)
and
[𝑁 𝑝 , 𝐴†𝑞 ] = [𝐼 𝑝 − 𝑍 𝑝 , 𝑄 −𝑞 ]
1
= [𝐼 𝑝 − 𝑍 𝑝 , 𝑋𝑞 − 𝑖𝑌𝑞 ]
2
1
= ( [𝐼 𝑝 , 𝑋𝑞 ] − 𝑖[𝐼 𝑝 , 𝑌𝑞 ] − [𝑍 𝑝 , 𝑋𝑞 ] + 𝑖[𝑍 𝑝 , 𝑌𝑞 ] )
2
52
𝑋 𝑝 − 𝑖𝑌𝑝
= 𝛿 𝑝𝑞
2
= 𝛿 𝑝𝑞 𝐴†𝑝 , (3.78)
and
[𝑁 𝑝 , 𝐴𝑞 ] = [𝐼 𝑝 − 𝑍 𝑝 , 𝑄 +𝑞 ]
1
= [𝐼 𝑝 − 𝑍 𝑝 , 𝑋𝑞 + 𝑖𝑌𝑞 ]
2
1
= ( [𝐼 𝑝 , 𝑋𝑞 ] + 𝑖[𝐼 𝑝 , 𝑌𝑞 ] − [𝑍 𝑝 , 𝑋𝑞 ] − 𝑖[𝑍 𝑝 , 𝑌𝑞 ] )
2
𝑋 𝑝 + 𝑖𝑌𝑝
= 𝛿 𝑝𝑞
2
= 𝛿 𝑝𝑞 𝐴 𝑝 . (3.79)
53
CHAPTER 4
LIPKIN MODEL
4.1 Introduction
The Lipkin model is an exactly solvable, many-body toy-model, first introduced in
1965 by Lipkin, Meschkov, and Glick [56]. It is often used to test the validity of nuclear
many-body methods. The version we consider here describes pairing interactions between
two levels (with the same 𝑗-value that straddle the Fermi level). The model consists of 𝑁
nucleons, distributed over two, Ω-degenerate levels which are indexed by 𝜎 = ±1; here
Ω = 2 𝑗 + 1. The model is depicted schematically in Figure 4.1 below. The solid lines
represent the available energy levels while the dashed line represents the Fermi-level. The
model is described by the following Hamiltonian
1 Õ † 1 Õ † †
𝐻= 𝜖 𝜎𝑎 𝑛𝜎 𝑎 𝑛𝜎 − 𝑉 𝑎 𝑎 𝑎 𝑚 𝜎¯ 𝑎 𝑛𝜎¯ , (4.1)
2 𝑛𝜎 2 𝑛𝑚𝜎 𝑛𝜎 𝑚𝜎
where 𝑛, 𝑚 = 1, 2, ..., Ω and 𝜎 = ±1 (with 𝜎 ¯ = −𝜎). The single-particle energy 𝑒 is the
amount of energy required to move a nucleon between the lower-level (𝜎 = −1) which
has energy −𝜖/2 to the upper-level (𝜎 = +1) which has energy 𝜖/2. Additionally, the
interaction strength 𝑉 is the energy required to move a pair of nucleons between the lower
and upper levels. Note that in this model, nucleons must move between levels in pairs,
either two in the lower level moving together to the upper or vice verse; this is why it is
Figure 4.1: Schematic of Lipkin Model.
54
described as a pairing model.
4.2 Classical Solutions
4.2.1 Full Configuration Interaction
The Lipkin model can be exactly solved via the full configuration interaction (FCI)
method described in subsection (2.4.1). The FCI basis consists of the Slater determinants
|𝑛i = |𝑛1 · · · 𝑛2Ω i = (𝑎 †1 ) 𝑛1 ...(𝑎 †2Ω ) 𝑛2Ω |0i , (4.2)
where 𝑛 𝑘 = 0, 1 is the occupation number of the state with index 𝑚 = b𝑘/2c and 𝜎 = 2[𝑘
(mod 2)] − 1, for 𝑘 = 1, 2, ..., 2Ω. That is, even (odd) values of 𝑘 index the upper (lower)
level, respectively. For 𝑁 nucleons, the FCI basis consists of all states |𝑛i with a Hamming
weight of 𝑁; that is, 𝑛 𝑘 = 1 for 𝑁 values of 𝑘. The size of the basis is thus 2Ω 𝑁 . The
diagonal Hamiltonian matrix elements are given by
𝜖
h𝑚 1 · · · 𝑚 2Ω |𝐻|𝑛1 · · · 𝑛2Ω i = (𝑁+ − 𝑁− ) , (4.3)
2
where 𝑛 𝑘 = 𝑚 𝑘 for all 𝑘. Here
Ω−1
Õ
𝑁+ = 𝑛2𝑘+1 , (4.4)
𝑘=0
Ω−1
Õ
𝑁− = 𝑛2𝑘 , (4.5)
𝑘=0
are the number of nucleons in the upper and lower levels, respectively. The off-diagonal
matrix elements are given by
𝑉
h𝑚 1 · · · 𝑚 2Ω |𝐻|𝑛1 · · · 𝑛2Ω i = − , (4.6)
2
where 𝑛2𝑘 𝑛2 𝑗 𝑛2𝑘+1 𝑛2 𝑗+1 = 𝑥𝑥 𝑥¯ 𝑥¯ and 𝑚 2𝑘 𝑚 2 𝑗 𝑚 2𝑘+1 𝑚 2 𝑗+1 = 𝑥¯ 𝑥𝑥𝑥¯ for exactly one pair (𝑘, 𝑗)
from 𝑘 ≠ 𝑗 = 0, ..., Ω − 1, and 𝑛𝑙 = 𝑚 𝑙 for all 𝑙 ≠ 2𝑘, 2𝑘 + 1, 2 𝑗, 2 𝑗 + 1. Here, 𝑥 = 0, 1
55
with 0̄ = 1 and 1̄ = 0. The eigenvalues and eigenvectors are then found through direct
diagonalization of the Hamiltonian matrix 𝐻 which has elements
𝐻𝑛1 ...𝑛2Ω ,𝑛1 ...𝑛2Ω = h𝑚 1 · · · 𝑚 2Ω |𝐻|𝑛1 · · · 𝑛2Ω i . (4.7)
The eigenvalue energies are computed for the case Ω = 𝑁 = 4 against various values of the
interaction strength 𝑉. They are depicted as lines in Figure 4.2. This was the largest value
of Ω that could be solved in a reasonable amount of time on my laptop, underscoring the
exponential time-scaling of the FCI method.
Figure 4.2: The energy eigenvalues (𝐸) of the Lipkin model, are plotted for various
interaction strengths (𝑉). The level degeneracy Ω and particle number 𝑁 are both four
while the single-particle energy 𝜖 is one. The different energies 𝑒 𝑘 for 𝑘 = 0, 1, ..., 6 are
depicted by different colors, labeled in the plot itself. The solid lines are the results of the
FCI method while the dots are the results of the symmetry method.
56
4.2.2 Symmetry Method
Following the procedure of the symmetry method laid out in subsection (2.4.2), we
start by identifying the symmetries of the Lipkin Hamiltonian. The first symmetry is the
particle-number. The particle number operator
Õ
𝑁= 𝑎 †𝑛𝜎 𝑎 𝑛𝜎 , (4.8)
𝑛𝜎
commutes with the Lipkin Hamiltonian. This can be seen by examining the Hamiltonian
(4.1) and noticing that the one-body part simply counts particles while the two-body term
moves particles in pairs. Thus, the Hamiltonian conserves particle number. To find more
symmetries, we rewrite the Lipkin Hamiltonian in terms of SU(2) operators
1
𝐻 = 𝜖 𝐽𝑧 + 𝑉 (𝐽+2 + 𝐽−2 ), (4.9)
2
via the mapping
Õ
𝐽𝑧 = 𝑗 𝑧(𝑛) , (4.10)
𝑛
Õ
𝐽± = 𝑗±(𝑛) , (4.11)
𝑛
where
1Õ †
𝑗 𝑧(𝑛) = 𝜎𝑎 𝑛𝜎 𝑎 𝑛𝜎 , (4.12)
2 𝜎
𝑗±(𝑛) = 𝑎 †𝑛± 𝑎 𝑛∓ . (4.13)
These operators obey the SU(2) commutation relations
[𝐽+ , 𝐽− ] = 2𝐽𝑧 , (4.14)
[𝐽𝑧 , 𝐽± ] = ±𝐽± , (4.15)
57
as justified in Appendix A. Here, the ladder operators are defined as 𝐽± = 𝐽𝑥 ± 𝑖𝐽𝑦 . With
this rewriting, we can see that the total spin operator 𝐽 2 , which is defined as
1
𝐽 2 = 𝐽𝑥2 + 𝐽𝑦2 + 𝐽𝑧2 = {𝐽+ , 𝐽− } + 𝐽𝑧2 , (4.16)
2
commutes with the Hamiltonian since the Hamiltonian is written explicitly in terms of
SU(2) operators and 𝐽 2 is the center of SU(2), meaning that it commutes with all the
group’s elements. Finally, we note that the signature operator
𝑅 = 𝑒𝑖𝜋𝐽𝑧 , (4.17)
commutes with the Hamiltonian, which can be explained as follows: Writing 𝐽𝑧 as
1
𝐽𝑧 = (𝑁+ − 𝑁− ), (4.18)
2
𝑎 †𝑛± 𝑎 𝑛± , allows us to see that it measures half the difference between the
Í
where 𝑁± = 𝑛±
number of particles in the upper and lower levels. Thus, the possible eigenvalues 𝑟 of the
signature operator are
+1, 𝑗 𝑧 = 2𝑛
1
+𝑖, 𝑗 𝑧 = 2𝑛 +
2
𝑟= (4.19)
−1, 𝑗 𝑧 = 2𝑛 + 1
3
−𝑖, 𝑗 𝑧 = 2𝑛 +
2
for 𝑛 ∈ Z. Note that 𝑟 is real or imaginary if the number of particles 𝑁 is even or odd,
respectively. Since, as discussed above, the Lipkin Hamiltonian conserves 𝑁, 𝑟 cannot
jump between being real and imaginary. Additionally, because particles must be moved in
58
pairs, and 𝐽𝑧 measures half the difference between particles in the upper and lower levels,
𝑗 𝑧 can only change by as
1
𝑗𝑧 → [(𝑁+ ± 2𝑛) − (𝑁− ∓ 2𝑛)]
2
= 𝐽𝑧 ± 2𝑛. (4.20)
We have determined that the symmetry operators 𝑁, 𝐽, and 𝑅 commute with the Hamilto-
nian. Let their eigenvalues be 𝑛, 𝑗, and 𝑟, respectively. These become our new quantum
numbers. Starting with the particle number operator 𝑁, because 𝐻 cannot mix states
from one particle number to another ( h𝑁 |𝐻|𝑁 0i = 0 if 𝑁 ≠ 𝑁 0) the Hamiltonian matrix
is block diagonal with 𝑁 blocks. Each block corresponds to a different particle number
(𝑛 = 0, 1, ..., 2Ω) with size 𝑛2 /4+ 𝑛. This is a direct result of the particle number symmetry.
We now move on to the total-spin operator. Because the Hamiltonian cannot mix states
with different 𝐽, we have that h𝐽 |𝐻|𝐽 0i = 0 for 𝐽 ≠ 𝐽 0. Now, for a given 𝑁, we label our
basis states | 𝑗 𝑗 𝑧 i where 𝑗 𝑧 = − 𝑗, − 𝑗 + 1, ..., 𝑗 − 1, 𝑗. The non-zero Hamiltonian matrix
elements are
h𝐽𝐽𝑧 | 𝐻 𝐽 0 𝐽𝑧0 = 𝛿 𝐽𝐽 0 𝜖 𝑗 𝑧 , (4.21)
𝑉p
h𝐽𝐽𝑧 + 2| 𝐻 |𝐽𝐽𝑧 i = − [ 𝑗 ( 𝑗 + 1) − 𝑗 𝑧 ( 𝑗 𝑧 + 1)]
2
p
× [ 𝑗 ( 𝑗 + 1) − ( 𝑗 𝑧 + 1)( 𝑗 𝑧 + 2)], (4.22)
𝑉p
h𝐽𝐽𝑧 − 2| 𝐻 |𝐽𝐽𝑧 i = − [ 𝑗 ( 𝑗 + 1) − 𝑗 𝑧 ( 𝑗 𝑧 − 1)]
2
p
× [ 𝑗 ( 𝑗 + 1) − ( 𝑗 𝑧 − 1)( 𝑗 𝑧 − 2)], (4.23)
since the operators that make up the Hamiltonian act on the basis states as follows:
𝐽𝑧 |𝐽𝐽𝑧 i = 𝑗 𝑧 |𝐽𝐽𝑧 i , (4.24)
59
p
𝐽± |𝐽𝐽𝑧 i = 𝑗 ( 𝑗 + 1) − 𝑗 𝑧 ( 𝑗 𝑧 ± 1) |𝐽𝐽𝑧 ± 1i , (4.25)
Note that the maximum possible value of 𝑗 𝑧 is 𝑁/2 which would correspond to the state
where all 𝑁 particles are spin up:
1Õ 𝑁
𝑗 𝑧 |↑ · · · ↑i = 𝑁𝑚𝜎 |↑ · · · ↑i = . (4.26)
2 𝑚𝜎 2
Therefore, the maximum value of 𝑗 is also 𝑁/2 and thus its possible values are 𝑗 =
𝑁/2, 𝑁/2 − 1, ..., 1 if 𝑁 is even and 𝑗 = 𝑁/2, 𝑁/2 − 1, ..., 1/2 if 𝑁 is odd. For each 𝑗,
the possible values of 𝑗 𝑧 are 𝑗 𝑧 = 𝑗, 𝑗 − 1, ..., − 𝑗. Thus, there are 2 𝑗 + 1 possible values
of 𝑗 𝑧 for each 𝑗. This implies that each 𝑛-block of the Hamiltonian matrix is itself a block
diagonal matrix consisting of b𝑁/2c blocks. Each block corresponds with a total spin
value ( 𝑗 = 𝑁/2, 𝑁/2 − 1, ...) and has length 2 𝑗 + 1. This is the direct result of the total
spin symmetry. Finally, we move to the signature 𝑅. Each 𝑗-block is again, itself a block
diagonal with two blocks (𝑟 = ±1 if 𝑁 is even or 𝑟 = ±𝑖 if 𝑁 is odd) which have size 𝑗 and
𝑗 + 1, respectively. The energies are computed by direct diagonalization of the Hamiltonian
matrix we’ve been describing above.
One can see in Figure 4.2 that the energies computed via the symmetry method (depicted
as block dots) exactly match the results of the FCI method. This is a demonstration that
the symmetry method is indeed a valid, exact solution. The symmetry method is also used
to solve the Lipkin Hamiltonian for the case of Ω = 𝑁 = 10. With the FCI method, this
would involve diagonalizing a size 20 5
10 ∼ 10 , square matrix. But, with the symmetry
method, one need only diagonalize several smaller square matrices, the largest of which
is size 30, which is 𝑗 ( 𝑗 + 1) for 𝑗 = 𝑁/2. The eigenvalues of the Lipkin Hamiltonian for
single-particle energy 𝜖 = 1 are plotted against various pairing strengths 𝑉 in Figure 4.3.
60
Figure 4.3: The energy eigenvalues (𝐸) of the Lipkin model, computed via the symmetry
method, are plotted for various interaction strengths (𝑉). The level degeneracy Ω and
particle number 𝑁 are both ten while the single-particle energy 𝜖 is one. The solid and
dashed lines correspond to signature numbers 𝑟 = +1 and 𝑟 = −1, respectively. The colors
yellow, magenta, cyan, red, green, and blue correspond to 𝑗 = 0, 1, ..., 5, respectively.
Each color represents a different value of 𝑗. For example, the eigenvalue energies
plotted in blue were the result of diagonalizing the 𝑗 = 5 block. The lines are solid and
dashed to correspond to the signature 𝑟 = +1 and 𝑟 = −1, respectively. We note that for
𝑉 = 0, there are 11 = 𝑁 + 1 values that the energies can take, corresponding to the fact
that the Lipkin model with no interaction strength is simply 𝜖 𝐽𝑧 ; thus the Hamiltonian
simply counts half the difference between the number of particles in the two levels, a
number which has 11 possible values: 𝑗 𝑧 = 0, ±1, ±2, ±3, ±4, ±5. However, as the pairing
strength is turned on, the energies start to bend and split. We notice that as 𝑉 increase, the
energies for 𝑟 = +1 and 𝑟 = −1 start to pair up and equal one another, their states becoming
degenerate.
61
4.2.3 Hartree-Fock Method
As mentioned in subsection 2.4.3, the ansatz we use for the variational method is the
particle-number conserving product state which, for the Lipkin model, is labeled as
! 𝑁
Õ Ö
|𝜏i = exp 𝜏𝑛+ 𝑛− 𝑎 †𝑛+ 𝑎 𝑛− 𝑎 †𝑘 |0i , (4.27)
𝑛 𝑘=1
where 𝜏𝑛+ 𝑛− is a variational parameter. This is motivated by Thouless’s theorem which
states that such an ansatz can rotate any Slater determinant into any other. Since we are only
considering single Slater determinants, this is exactly what we desire. Here, we consider
the half-filled case 𝑁 = Ω. Thus, we can start the ansatz in the state
Ö 𝑁
𝑎 †𝑘 |0i = |𝐽 − 𝐽i . (4.28)
𝑘=1
However, because the Lipkin Hamiltonian treats all (𝑛+, 𝑛−) pairs equivalently (the two-
body coefficient 𝑉 is independent of 𝑛 and 𝜎) we can set
𝜏𝑛+ 𝑛− = 𝜏, (4.29)
a new variational parameter, for all 𝑛. With this, the normalized ansatz becomes
|𝜏i = (1 + |𝜏| 2 ) 𝐽 𝑒 𝜏𝐽+ |0i , (4.30)
where the normalization is determined in Appendix B. We now calculate the expectation
value of the Hamiltonian in the ansatz. Using Appendix C, we derive
Ω |𝜏| 2 − 1 𝜏 2 + 𝜏¯ 2
𝐸 (𝜏) = h𝜏|𝐻|𝜏i = − 𝑉 (Ω − 1)
𝜖 2 2 ,
2 |𝜏| + 1 2
|𝜏| + 1
1 𝜒 2
= − 𝜖Ω cos 𝜃 + sin 𝜃 cos 2𝜙 , (4.31)
2 2
62
Figure 4.4: The expectation value of the Lipkin Hamiltonian in the SU(2) coherent state
ansatz (4.30) is plotted vs theta for various values of 𝜒 which are distinguished by different
colors, labeled on the plot itself. The black dots represent the minimum energy (4.38).
where we’ve defined the coupling strength 𝜒 to be
𝑉
𝜒= (Ω − 1). (4.32)
𝜖
This energy profile 𝐸 (𝜏) = h𝜏|𝐻|𝜏i is plotted for various values of 𝜒 in Figure 4.4. The
plot shows the symmetry breaking that occurs when 𝜒 becomes greater than 1. For 𝜒 ≤ 1,
there is only one 𝜃 min (namely 𝜃 min = 0). However, for 𝜒 > 1, there exists two values of
𝜃 min which give the same, correct ground-state energy, and are symmetric about 𝜃 = 0. To
minimize 𝐸 (𝜏), we first set its derivatives to zero, resulting in the following expressions:
𝜕𝐸 1
0= = 𝜖Ω sin 𝜃 (1 − 𝜒 cos 𝜃 cos 2𝜙), (4.33)
𝜕𝜃 2
𝜕𝐸 1
0= = 𝜖Ω𝜒 sin2 𝜃 sin 2𝜙, (4.34)
𝜕𝜙 2
the first of which implies either 𝜃 min = 0, 𝜋 or cos 𝜃 min cos 2𝜙min = 1/𝜒 and the second
of which implies either 𝜃 min = 0, 𝜋 or 𝜙min = 0, 𝜋/2. Second, we demand that its second
derivatives are positive
𝜕2 𝐸 1
0< = 𝜖Ω(cos 𝜃 − 𝜒 cos 2𝜃 cos 2𝜙), (4.35)
𝜕𝜃 2 2
63
𝜕2 𝐸
0< 2
= 𝜖Ω𝜒 sin2 𝜃 cos 2𝜙, (4.36)
𝜕𝜙
which implies (assuming, without loss of generality, that 𝜒 > 0) −𝜋/3 < 𝜃 min < 𝜋/3 and
𝜙min < 𝜋/4. And third, we set the cross-derivative equal to zero
𝜕2 𝐸
0= = 𝜖Ω𝜒 cos 2𝜃 sin 2𝜙, (4.37)
𝜕𝜃𝜕𝜙
which implies 𝜃 min = 0, 𝜋. Combining all these conditions implies
if 0 < 𝜒 ≤ 1
0,
𝜃 min = , (4.38)
cos−1 𝜒 ,
1
if 1 < 𝜒
𝜙min = 0. (4.39)
The minimum variational parameter is thus, 𝜏min = 𝜏(𝜃 min , 𝜙min ) where 𝜏(𝜃, 𝜙) = tan(𝜃/2)𝑒 −𝑖𝜙 ,
as defined in Appendix C. Finally, we plug these minimum parameters (4.38 and 4.39)
back into 𝐸 (𝜏) (4.31) to find the Hartree-Fock ground state energy to be
− 2𝜖 Ω,
if 0 < 𝜒 < 1
𝐸0 = (4.40)
𝜖 1
− 4 Ω 𝜒 + 𝜒 ,
if 1 < 𝜒.
The Hartree-Fock method is bench-marked against the exact answer in Figure 4.5. The
Hartree-Fock calculated ground-state energy (4.40) is plotted against various values of the
coupling strength 𝜒. Alongside it we plot the exact ground state energy computed using
the symmetry method. We more precisely inspect the performance of the Hartree-Fock
method by plotting the relative error between the Hartree-Fock and exact energies in Figure
4.6. We note that the two methods start in exact agreement at 𝜒 = 0. The magnitude of
the relative error increases until just after 1. It then decreases as 𝜒 increases past 1 and
64
Figure 4.5: The Hartree-Fock and exact energies of the Lipkin are plotted against various
values of 𝜒.
Figure 4.6: The relative difference between the Hartree-Fock and exact energies is plotted
against various values of 𝜒.
seems to asymptotically approach a small error. Thus, one could say that the Hartree-Fock
method is a good approximation for either very small or very large 𝜒.
4.3 Quantum Solutions
To solve the Lipkin model with a quantum computer, the first step is to map the system
to a set of qubits. We’ll restrict ourselves here to the half-filled case where the number of
particles 𝑁 equals the degeneracy of the states Ω. One could assign each possible state
65
(𝑛, 𝜎) to a qubit such that the qubit being in the state |1i or |0i would imply that the state
(𝑛, 𝜎) is occupied or unoccupied, respectively. This mapping scheme (which we’ll call
occupation mapping) requires 2Ω qubits. Additionally, any ansatz that would restrict the
minimization search to the correct subspace of constant Hamming weight 𝑁 (since the
number of particles 𝑁 is conserved) would necessitate the use of at least four-qubit gates.
This is because moving a pair of particles in this scheme would require two annihilation
operators on the states from which the pair particles move and two creation operators on
the states to which the pair of particles move. That is, it takes a four-qubit gate to change
between the states |1100i and |0011i, for example. And, as discussed in the chapter of
quantum computing, it is only known how to efficiently decompose up to two-qubit gates.
Thus the involvement of four qubit gates would necessitate a longer depth circuit than one
involving only two and one qubit gates, creating more noise and less accurate results.
However, because there are only two energy levels in the Lipkin model, another natural
mapping is possible. In this mapping scheme (which we’ll call level mapping) each doublet
((𝑛, +1), (𝑛, −1)) would be assigned a qubit such that the qubit being in the state |0i or |1i
would imply that the particle is in the (𝑛, +1) or (𝑛, −1) state, respectively. Note that these
are the only two possible configurations of the doublet as we are restricting ourselves to the
half-filled case and the Lipkin Hamiltonian only moves particles between energy levels, not
degenerate states. Thus the level mapping only requires Ω qubits which is half that of the
occupation mapping. Additionally, any ansatz that would restrict the minimization search
to the correct subspace of constant Hamming weight 𝑁 requires at most, only two-qubit
gates. This is because moving a pair of particles in this scheme only changes the state
of two doublets (and therefore qubits). That is, it only takes a two-qubit gate to change
66
between the states |00i and |11i, for example. As an efficient decomposition two-qubit
gates is known, the ansatz for this mapping would be shorter (and thus less noisy) than that
of the previous mapping.
One could imagine a third mapping scheme which would require even less qubits in
which each of the possible states in the spin basis |𝐽𝐽𝑧 i is mapped to a single qubit. In
this spin mapping, there are only 2𝐽 + 1 possible states (since 𝐽𝑧 = −𝐽, −𝐽 + 1, ..., 𝐽 − 1, 𝐽)
for each value of 𝐽. And, since the Hamiltonian is block diagonal (with a different block
for each 𝐽) the eigenvalues of the Hamiltonian are simply the eigenvalues of each block,
which may be calculated separately. Since the maximum value of 𝐽 is 𝐽max = 𝑁/2, the
largest number of qubits would be 2𝐽max + 1 = 𝑁 + 1. However, b𝑁/2c different circuit
would need to be used for minimization for all possible values of 𝐽, to explore the entire
Hilbert space. (The minimum of the set of minimum energies that each circuit finds would
be the ground state energy of the entire system.) This increases, linearly, the amount of
time required to find the ground state energy.
After reviewing the three possible mappings, it is our view that the level mapping [17]
is the best suited for NISQ era devices given its low qubit count and ability to search the
entire relevant Hilbert space with one circuit (which reduces time to solution) and the fact
that at most, only two-qubit gates are required of the ansatz, leading to shorter depth (and
thus less noisy) circuits. With this mapping, the Hamiltonian takes the form utilized in the
symmetry method (subsection 4.2.2) which was given by equation (4.9) as
1
𝐻 = 𝜖 𝐽𝑧 + 𝑉 (𝐽+2 + 𝐽−2 ). (4.41)
2
Plugging the mapping from the total 𝐽 operators to individual 𝑗 operators (equations 4.12
67
and 4.13) yields
Õ !2 ! 2
Õ 1 (𝑛)
Õ
𝑗 𝑧(𝑛) (𝑛)
𝐻=𝜖 + 𝑉 𝑗 + + 𝑗 − (4.42)
𝑛
2
𝑛 𝑛
Õ 1 Õ
=𝜖 𝑗 𝑧(𝑛) + 𝑉 𝑗+(𝑛) 𝑗+(𝑚) + 𝑗 −(𝑛) 𝑗−(𝑚) (4.43)
𝑛
2 𝑛,𝑚
Õ Õ
=𝜖 𝑗 𝑧(𝑛) + 2𝑉 (𝑛) (𝑚)
𝑗𝑥 𝑗𝑥 − 𝑗 𝑦 𝑗 𝑦 (𝑛) (𝑚)
, (4.44)
𝑛 𝑛<𝑚
where we’ve used the definitions
𝑗±(𝑛) = 𝑗𝑥(𝑛) ± 𝑖 𝑗 𝑦(𝑛) . (4.45)
To convert to Pauli matrices, we’ll make the transformations
𝑗𝑥(𝑛) → 𝑋𝑛 /2, (4.46)
𝑗 𝑦(𝑛) → 𝑌𝑛 /2, (4.47)
𝑗 𝑧(𝑛) → 𝑍𝑛 /2, (4.48)
which preserves the SU(2) commutation relations (4.14 and 4.15) and thus is allowable.
This transforms our Hamiltonian into
𝑛 𝑁
1 Õ 1 Õ
𝐻= 𝜖 𝑍𝑘 + 𝑉 (𝑋 𝑘 𝑋 𝑗 − 𝑌𝑘 𝑌 𝑗 ). (4.49)
2 𝑘=1 2 𝑖≠ 𝑗=1
With this form, we can clearly see that the first (one-body) term in the Hamiltonian returns
the energy −𝜖/2 or +𝜖/2 if the qubit representing the particle of a doublet is in the ground
(|1i) or excited (|0i) state, respectively. The action of the second (two-body) term in the
Hamiltonian can be determined by noting that
1
(𝑋 𝑋 − 𝑌𝑌 ) |00i = |11i , (4.50)
2
68
1
(𝑋 𝑋 − 𝑌𝑌 ) |01i = 0, (4.51)
2
1
(𝑋 𝑋 − 𝑌𝑌 ) |10i = 0, (4.52)
2
1
(𝑋 𝑋 − 𝑌𝑌 ) |11i = |00i . (4.53)
2
That is, the two-body term moves a pair of particles between the ground states |00i and the
excited states |11i of their respective doublets.
To construct an efficient ansatz, we must determine the subspace within which the
Hamiltonian lives. To begin, note that particles are only ever moved between energy levels
in pairs. This implies that all possible states have a Hamming weight of constant parity
(odd or even); this is the same as the signature 𝑟 being conserved. Further, note that the
Hamiltonian’s coefficients (𝜖 and 𝑉) are state independent (do not depend on the indices 𝑛
or 𝑚) as the states labeled by these indices are degenerate and thus have the same energy
level. Thus, the Hamiltonian treats all states with the same number of excited particles
(Hamming weight of the state) as the same. Therefore, the following ansatz forms exactly
cover the subspace within which the 𝑁-degenerate Hamiltonian explores:
b𝑛/2c
Õ
𝑛
|𝜓even i = 𝑐 2𝑘 𝐷 2𝑘 , (4.54)
𝑘=0
b𝑛/2c
Õ
𝑛
|𝜓odd i = 𝑐 2𝑘+1 𝐷 2𝑘+1 . (4.55)
𝑘=0
Here 𝐷 𝑛𝑘 represents a Dicke state which is defined as equal superposition of all 𝑛-qubit
states with Hamming weight 𝑘. That is
1 Õ
𝐷 𝑛𝑘 = q |𝑥i , (4.56)
𝑛 𝑛
𝑘 𝑥∈ℎ 𝑘
69
where ℎ𝑛𝑘 = {|𝑥i | l(𝑥) = 𝑛, wt(𝑥) = 𝑘 }. One way to prepare these ansatzes is to do so
exactly as it is known how to deterministically prepare Dicke states with linear depth [9].
The reference provides an algorithm for preparing a set of gates 𝑈 𝑘𝑛 that prepares a Dicke
state from a product state of Hamming weight 𝑘; that is
𝑈 𝑘𝑛 |1i ⊗𝑘 |0i ⊗𝑛−𝑘 = 𝐷 𝑛𝑘 . (4.57)
It then describes how to one can create an arbitrary superposition of Dicke states, which
we modify here to restrict ourselves to a Hamming weight of constant parity. The circuit
to construct such a state (for the 𝑘 = 6 case, as an example) is given below
|0i 𝑅 𝑦 (𝜃 0 ) • 𝑅𝑧 (𝜙0 ) (4.58)
|0i •
|0i 𝑅 𝑦 (𝜃 1 ) • 𝑅𝑧 (𝜙1 )
𝑈 𝑘𝑛
|0i •
|0i 𝑅 𝑦 (𝜃 2 ) • 𝑅𝑧 (𝜙2 )
|0i
The 𝑅 𝑦 gates and CNOT gates prepare an arbitrary real superposition of product states with
even Hamming weight 𝑘; then the 𝑅𝑧 gates add arbitrary phases to each of the states
|000000i → cos(𝜃 0 /2) |000000i
+ sin(𝜃 0 /2) cos(𝜃 1 /2)𝑒𝑖𝜃 0 |110000i
+ sin(𝜃 0 /2) sin(𝜃 1 /2) cos(𝜃 2 /2)𝑒𝑖(𝜃 0 +𝜃 1 ) |111100i
+ sin(𝜃 0 /2) sin(𝜃 1 /2) sin 𝜃 2 /2)𝑒𝑖(𝜃 0 +𝜃 1 +𝜃 2 ) |111111i . (4.59)
70
Finally, 𝑈 𝑘𝑛 converts each product state to its corresponding Dicke state. Thus, all together
the circuit acts as
|000000i → cos(𝜃 0 /2) 𝐷 60
+ sin(𝜃 0 /2) cos(𝜃 1 /2)𝑒𝑖𝜃 0 𝐷 62
+ sin(𝜃 0 /2) sin(𝜃 1 /2) cos(𝜃 2 /2)𝑒𝑖(𝜃 0 +𝜃 1 ) 𝐷 64
+ sin(𝜃 0 /2) sin(𝜃 1 /2) sin 𝜃 2 /2)𝑒𝑖(𝜃 0 +𝜃 1 +𝜃 2 ) 𝐷 66 . (4.60)
The circuit (4.58) can be extended naturally for any even value of 𝑘. For odd values of
𝑘, one need simply add a single-qubit to the top of the circuit for 𝑘 − 1 and apply the
𝑋 gate to it. Although this ansatz has linear depth, the circuit for 𝑈 𝑘𝑛 involves several
double-controlled gates which involve the usage of several CNOT gates to decompose. As
the CNOT gate is often the noisiest gate in NISQ era quantum computers, it is best to
minimize their use.
4.4 Results
In this section, we test out ansatz 4.58 for the Lipkin model with parameters. Ω = 4,
𝑒 = 1 and 𝑣 = 1. One can see if Figure 4.7 that running VQE with our ansatz matches
the exact energy for the most part, and always performs better than Hartree-Fock. Because
the simulations of VQE were noiseless, we hypothesize that the slight variations in some
of the VQE dots (red) off of the exact energy line (blue) could be due to the minimization
algorithm failing to converge properly. Finding a set of initial parameters that would
initialize us to a state with a large overlap with the ground state would be beneficial.
71
Figure 4.7: Comparison of energies for the Lipkin model calculated through direct diago-
nalization (Exact), Hartree-Fock (HF), and the variational quantum eigensolver (VQE).
4.5 Conclusion
In this section, we introduced the Lipkin model which serves as a toy model in nuclear
physics with which to benchmark new techniques. We first solve the problem through var-
ious classical avenues including the full configuration interaction, symmetry, and Hartree-
Fock methods. We then discussed the different ways to map the problem from its fermionic
space to the spin space with which which quantum computers deal. We gave a novel way
to construct one form of the ansatz for the model, implemented it as the ansatz for VQE
and compared the quantum results to the classical results. This section served as a first
example of how quantum computers can be used to solve a toy nuclear pairing model.
72
CHAPTER 5
PAIRING MODEL
5.1 Introduction
There exists an intriguing phenomenon in physics called pairing in which fermions
"pair up", meaning that they move close together in space and tend to move energy levels
together. The phenomenon of nuclear pairing can be understood from a simple symmetry
argument. Since nucleons are fermions, the overall wavefunction of a pair nucleons
must be anti-symmetric. Any such wavefunction can be written as the product of three
separable wavefunctions: a spatial function, a spin function, and an isospin function.
Pairing occurs between two fermions of opposite spin, which necessitates that the spin
(𝑆 = 0) function of their combined system be anti-symmetric. Additionally, pairing occurs
most strongly between two nucleons of the same type (proton-proton or neutron-neutron)
which is described by an isospin of 𝑇 = 1, necessitating that the isospin function of
their combined system be symmetric. The spin function being anti-symmetric and the
isospin function being symmetric implies that the spatial function must be symmetric (to
preserve the overall anti-symmetry of the pair). This occurs when the angular momentum
component of the spatial wavefunction is zero, 𝑙 = 0, resulting in a wavefunction whose
density has a strong peak near 𝑟 = 0, where 𝑟 is the separation between the two nucleons.
Thus, the two nucleons tend to stick close together and can be approximated as a pair that
moves together. Pairing occurs most strongly when 𝐽 = 0 because 𝐽 = 𝑆 + 𝐿 and we
determined above that 𝐿 = 𝑆 = 0. Because of this, a large energy gap is produced between
the 𝐽 = 0 and 𝐽 > 0 states by the nuclear force between identical nucleons. Therefore,
one may approximate such a system via the so-called pairing interaction which only acts
73
on the paired 𝐽 = 0 state [22]. As we are approximating the residual interaction between
identical nucleons, the pairing approximation is only suitable for semi-magic nuclei with
valence nucleons of a single type [47]. Indeed, this paring approximation has been used
to approximate such systems in heavy nuclei [27]. It will be seen that models that include
only pairing interactions in their two-body forces are computationally simpler to solve and
yet, as just mentioned, still applicable to real-world systems. Thus, such models serve as an
area of interest for applications of NISQ era quantum computing. The pairing interaction
was first introduced by Racah for atomic physics [65]. The algebra of identical nucleon
pairs (isospin 𝑡 = 0) is found to be isomorphic to SU(2) and is thus called quasi-spin.
This is advantages for quantum computers whose qubits live in SU(2). One of the earliest
applications of pairing was by Bardeen, Cooper, and Schrieffer in their famous model
(BCS) of superfluidity in condensed matter in 1957 [7]. The idea was adapted to pairing
in nuclei by Bohr, Mottelson, and Pines in 1958 [12].
In 1963, R.W. Richardson proposed a model consisting of fermions occupying non-
degenerate energy levels which interact solely through the pairing force ([1] and [66]). We
refer here to his model as the pairing model. It consists of 𝑃, non-degenerate energy levels,
occupied by 𝑁 pairs of fermions. Each pair consists of two fermions of opposite spin,
occupying the same energy level. It’s Hamiltonian is given by
Õ Õ
𝑑 𝑝 𝑎 †𝑝𝜎 𝑎 𝑝𝜎 + 𝑔𝑞 𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑞− 𝑎 𝑞+ .
𝑝
𝐻𝑝 = (5.1)
𝑝𝜎 𝑝𝑞
Here, the indices 𝑝 and 𝑞 sum over the set {0, ..., 𝑃 − 1}, representing the various energy
levels. Additionally, the index 𝜎 sums over the set {−, +}, representing the spin of each
fermion. The coefficient 𝑑 𝑝 represents the single particle energy corresponding to energy
𝑝
level 𝑝. The coefficients 𝑔𝑞 are the so-called pairing strengths which represent the energy
74
Figure 5.1: Example schematic of the paring model with 𝑃 = 4 energy levels and 𝑁 = 2
pairs of fermions. Shown are four energy levels with single-particle energies 𝑑0 , 𝑑1 , 𝑑2 , 𝑑3
of which the bottom two are initially filled by pairs of fermions. The dashed line represents
the Fermi level which divides the energy levels with single-particle energies 𝑑0 and 𝑑1 (the
hole states) from those with 𝑑2 and 𝑑3 (the particle states).
associated with moving a pair of fermions from the 𝑞 th to the 𝑝 th energy level. We will
consider various sets of pairing strengths in this section. An example of the model is
represented schematically in Figure 5.1.
To simplify the pairing model, its Hamiltonian can be rewritten in terms of pairing
operators as
Õ𝑛 Õ𝑛
𝑔𝑞 𝐴†𝑝 𝐴𝑞 .
𝑝
𝐻𝑝 = 𝑑𝑝 𝑁𝑝 + (5.2)
𝑝=1 𝑝,𝑞=1
Here, 𝑁 𝑝 is the pair number operator which counts the number of fermions occupying the
𝑝 th energy level. Furthermore, 𝐴†𝑝 and 𝐴 𝑝 are the pair fermionic creation and annihilation
operators, respectively, which create and annihilate pairs of fermions on the 𝑝 th energy
level. These operators are defined in terms of fermionic creation and annihilation operators
as follows
Õ
𝑁𝑝 = 𝑎 †𝑝𝜎 𝑎 𝑝𝜎 (5.3)
𝜎
𝐴†𝑝 = 𝑎 †𝑝+ 𝑎 †𝑝− (5.4)
𝐴 𝑝 = 𝑎 𝑝− 𝑎 𝑝+ , (5.5)
75
where 𝜎 sums over the set {+, −}. The purpose of this rewriting becomes clear once
one notices that these operators satisfy the SU(2) algebra described by the following
commutation relations
[ 𝐴 𝑝 , 𝐴†𝑞 ] = 𝛿 𝑝𝑞 (1 − 𝑁 𝑝 ), (5.6)
[𝑁 𝑝 , 𝐴†𝑞 ] = 2𝛿 𝑝𝑞 𝐴†𝑝 , (5.7)
[𝑁 𝑝 , 𝐴𝑞 ] = −2𝛿 𝑝𝑞 𝐴 𝑝 . (5.8)
which are proven in Appendix F.
5.2 Classical Solutions
Here we solve the pairing model via various classical techniques against which we will
benchmark our quantum techniques.
5.2.1 Exact Solution
If one restricts the pairing strength coefficients of the pairing model Hamiltonian (5.2)
to be constant
Õ 𝑃 Õ 𝑃
𝐻= 𝑑𝑝 𝑁𝑝 + 𝑔 𝐴†𝑝 𝐴𝑞 , (5.9)
𝑝=1 𝑝,𝑞=1
then there exists an exact solution, discovered by R.W. Richardson in 1963 [1]. The ansatz
that solves the model with 𝑁 pairs is given by
Ö𝑁
|Ψi = 𝐵†𝛼 |0i , (5.10)
𝛼=1
where
𝑃
Õ 1
𝐵†𝛼 = 𝐴† , (5.11)
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝜅
76
which, when plugged into the Schrodinger equation leads to a set of equations (the Richard-
son equations) which we shall re-derive here [46]. First, plugging the ansatz 5.10 into the
Schrodinger equation and rewriting, yields
𝐸 |Ψi = 𝐻 |Ψi
Ö 𝑁
=𝐻 𝐵†𝛼 |0i
𝛼=1
Ö𝑁
= [𝐻, 𝐵†𝛼 ] |0i , (5.12)
𝛼=1
since 𝐻 |0i = 0. The commutator can be expanded as follows
𝑁 Õ𝑁 𝛼−1 Ö 𝑁
Ö ©Ö
†ª † © †ª
[𝐻, 𝐵†𝛼 ] = 𝐵 𝛽 ® [𝐻, 𝐵𝛼 ] 𝐵𝛾 ® , (5.13)
𝛼=1 𝛼=1 « 𝛽=1 «𝛾=𝛼+1 ¬
¬
the inner commutator of which is given by
𝑃 𝑃 𝑃
Õ Õ Õ 1
[𝐻, 𝐵†𝛼 ] = [ 𝑑𝑝 𝑁𝑝 + 𝑔 𝐴†𝑝 𝐴𝑞 , 𝐴† ] (5.14)
𝑝=1 𝑝,𝑞=1 𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝜅
𝑃 𝑃 𝑃
Õ 1 ©Õ Õ
= †
𝑑 𝑝 [𝑁 𝑝 , 𝐴𝜅 ] + 𝑔 [ 𝐴†𝑝 𝐴𝑞 , 𝐴†𝜅 ] ®
ª
(5.15)
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝑝=1 𝑝,𝑞=1
« ¬
𝑃 𝑃 𝑃
Õ 1 ©Õ Õ
= 2𝛿 𝑝𝜅 𝑑 𝑝 𝐴†𝑝 + 𝑔 𝛿 𝑞𝜅 𝐴†𝑝 (1 − 𝑁 𝑞 ) ®
ª
(5.16)
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝑝=1 𝑝,𝑞=1
« ¬
𝑃 𝑃
Õ 1 Õ
= 2𝑑 𝜅 𝐴†𝜅 + 𝑔 𝐴†𝑝 (1 − 𝑁 𝜅 ) ®
© ª
(5.17)
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝑝=1
« ¬
𝑃 𝑃
Õ 𝐸𝛼 † 𝑔 Õ
†
=
2𝑑 𝜅 − 𝐸 𝛼 + 1 𝐴 𝜅 + 𝐴 𝑝 (1 − 𝑁 𝜅 ) (5.18)
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝑝=1
!
𝑃 𝑃
Õ Õ 1 − 𝑁𝜅
= 𝐸 𝛼 𝐵†𝛼 + 𝐴†𝑝 1 + 𝑔 . (5.19)
𝑝=1 𝜅=1
2𝑑 𝜅 − 𝐸 𝛼
77
Plugging this back into the Schrodinger equation (5.12) and applying it to the vacuum
yields
Ö 𝑁
𝐸 |Ψi = [𝐻, 𝐵†𝛼 ] |0i
𝛼=1
𝑁 𝛼−1 𝑃 𝑃
! 𝑁
−
© Ö † ª
Õ ©Ö
†ª
†
Õ
†
Õ 1 𝑁 𝜅
= 𝐵 𝛽 ® 𝐸 𝛼 𝐵𝛼 + 𝐴𝑝 1 + 𝑔 𝐵𝛾 ® |0i
𝛼=1 « 𝛽=1 𝑝=1 𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝛾=𝛼+1
¬ « ¬
= 𝐸 |Ψi
𝑁 𝑃
! 𝑃 𝑁
Õ Õ 𝑔 Õ © Ö
†ª
†
+ 1 + 𝐴 𝑝 𝐵 |0i
− 𝛽 ®
𝛼=1
𝜅=1
2𝑑 𝜅 𝐸 𝛼 𝑝=1
« 𝛽=1,𝛽≠𝛼 ¬
𝑁 𝛼−1 Õ 𝑃
! 𝑃 𝑁
Õ Ö
𝑔 Õ © Ö
†
† †ª
− 𝐵𝛾 ® |0i ,
© ª
𝐵𝛽® 𝑁𝜅 𝐴𝑝 (5.20)
𝛼=1 « 𝛽=1
2𝑑 𝜅 − 𝐸 𝛼
¬ 𝜅=1 𝑝=1
«𝛾=𝛼+1 ¬
Í𝑛
where we’ve defined 𝐸 = 𝛼=1 𝐸 𝛼 and used the definition of |Ψi (5.10) to simplify the first
term, which implies that the Schrodinger equation is satisfied if
𝑁 𝑃
! 𝑁
Õ
Õ 𝑔 Ö
†
1+ 𝐵 𝛽 |0i
𝛼=1
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝛽=1,𝛽≠𝛼
𝑁 𝛼−1 𝑃
! 𝑁
Õ ©Ö
†ª
Õ 𝑔 © Ö † ª
− 𝐵𝛽® 𝑁𝜅 𝐵𝛾 ® |0i = 0, (5.21)
2𝑑 𝜅 − 𝐸 𝛼
𝛼=1 « 𝛽=1 ¬ 𝜅=1 «𝛾=𝛼+1 ¬
where we’ve divided through by the constant term 𝑝 𝐴†𝑝 . Note that the second term from
Í
above can be re-written as
𝑁 𝛼−1 𝑃
! 𝑁
Õ ©Ö
†ª
Õ 𝑔 Ö
†
𝐵𝛽® [𝑁 𝜅 , 𝐵𝛾 ] |0i , (5.22)
𝛼=1 « 𝛽=1
2𝑑 𝜅 − 𝐸 𝛼
¬ 𝜅=1 𝛾=𝛼+1
since 𝑁 𝜅 |0i = 0. We’ll now expand the commutator from the above expression
© 𝛾−1 † ª
Ö𝑁 Õ 𝑁 Ö Ö 𝑁
† © †ª
[𝑁 𝜅 , 𝐵†𝛾 ] = 𝐵 𝜇 ® [𝑁 𝜅 , 𝐵𝛾 ] 𝐵𝜈 ® , (5.23)
𝛾=𝛼+1 𝛾=𝛼+1 « 𝜇=𝛼+1 «𝜈=𝛾+1 ¬
¬
78
the inner commutator of which is
𝑃
Õ 1
[𝑁 𝜅 , 𝐵†𝛾 ] = [𝑁 𝜅 , 𝐴𝜆† ] (5.24)
𝜆=1
2𝑑𝜆 − 𝐸 𝛾
𝑃
Õ 2
= 𝛿 𝜅𝜆 𝐴†𝜅 (5.25)
𝜆=1
2𝑑𝜆 − 𝐸 𝛾
2
= 𝐴† . (5.26)
2𝑑 𝜅 − 𝐸 𝛾 𝜅
Plugging this back into the second term (5.22) yields
𝑁 𝛼−1 𝑁 𝛾−1 𝑃
! 𝑁
Õ ©Ö
†ª
Õ
Ö
†ª
Õ 2𝑔 † ©
Ö
†ª
𝐵𝜈 ® |0i .
©
𝐵𝛽® 𝐵𝜇® 𝐴𝜅
(2𝑑 𝜅 − 𝐸 𝛼 )(2𝑑 𝜅 − 𝐸 𝛾 )
𝛼=1 « 𝛽=1 ¬ 𝛾=𝛼+1 « 𝜇=𝛼+1 ¬ 𝜅=1 «𝜈=𝛾+1 ¬
(5.27)
Applying partial fraction decomposition to the inner sum
1 1 1 1
= − , (5.28)
(2𝑑 𝜅 − 𝐸 𝛼 )(2𝑑 𝜅 − 𝐸 𝛾 ) 𝐸 𝛼 − 𝐸 𝛾 (2𝑑 𝜅 − 𝐸 𝛼 ) (2𝑑 𝜅 − 𝐸 𝛾 )
turns the second term (5.27) into
© 𝛾−1 † ª
𝑁 Ö Ö
© 𝛼−1 † ª Õ 𝑁 𝑁
Ö
Õ 2𝑔 † † †ª
(𝐵𝛼 − 𝐵𝛾 ) 𝐵𝜈 ® |0i ,
©
𝐵𝛽® 𝐵𝜇® (5.29)
𝐸 𝛼 − 𝐸 𝛾
𝛼=1 « 𝛽=1 ¬ 𝛾=𝛼+1 « 𝜇=𝛼+1 ¬ «𝜈=𝛾+1 ¬
which can be written as
𝑁 Õ 𝑁 Ö 𝑁
Õ
2𝑔 †
𝐵𝜈 ® |0i
© ª
𝐸𝛼 − 𝐸𝛾
𝛼=1 𝛾=𝛼+1 « 𝛽=1,𝛽≠𝛾 ¬
𝑁 Õ 𝑁 Ö 𝑁
Õ
2𝑔 †ª
− 𝐵𝜈 ® |0i .
©
(5.30)
𝐸𝛼 − 𝐸𝛾
𝛼=1 𝛾=𝛼+1 « 𝛽=1,𝛽≠𝛼 ¬
Switching the order of summation of the first term and swapping indices 𝛼 ↔ 𝛾 followed
by merging sums yields
𝑁 Õ𝛼−1 Ö 𝑁
Õ
2𝑔 †ª
− 𝐵𝜈 ® |0i
©
𝐸𝛼 − 𝐸𝛾
𝛼=1 𝛾=1 « 𝛽=1,𝛽≠𝛼 ¬
79
𝑁 Õ 𝑁 Ö 𝑁
Õ
2𝑔 †ª
− 𝐵𝜈 ® |0i
©
(5.31)
𝛼=1 𝛾=𝛼+1
𝐸 𝛼 − 𝐸 𝛾
« 𝛽=1,𝛽≠𝛼 ¬
𝑁 𝑁
Ö 𝑁
Õ Õ
2𝑔 †ª
=− 𝐵𝜈 ® |0i .
©
(5.32)
𝛼=1 𝛾=1,𝛾≠𝛼
𝐸 𝛼 − 𝐸 𝛾
« 𝛽=1,𝛽≠𝛼 ¬
Finally, plugging this back into the condition that satisfies the Schrodinger equation (5.21)
yields
𝑁 𝑃 𝑁 𝑁
Õ ©
Õ 𝑔 Õ 2𝑔 ª © Ö †ª
1 + + ® 𝐵 ® |0i = 0, (5.33)
𝛼=1 « 𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝛾=1,𝛾≠𝛼 𝐸 𝛼 − 𝐸 𝛾 𝛽=1,𝛽≠𝛼 𝛽
¬« ¬
which yields the Richardson equations
𝑃 𝑁
Õ 𝑔 Õ 2𝑔
1+ + = 0, (5.34)
𝜅=1
2𝑑 𝜅 − 𝐸 𝛼 𝛽=1,𝛽≠𝛼 𝐸 𝛼 − 𝐸 𝛽
where we’ve relabeled 𝛾 → 𝛽. This is a set of coupled, non-linear equations from which
one solves for the terms 𝐸 𝛼 and sums them to find the energy; recall
Õ𝑁
𝐸= 𝐸𝛼 . (5.35)
𝛼=1
However, the Richardson equations are notoriously difficult to solve, due to the presence
of singularities. Additionally, the pairing model can be solved exactly [60] if the single-
particle energies are degenerate (𝑑 𝑝 = 𝑑 for all 𝑝) and the pairing strength is separable
𝑝
(𝑔𝑞 = 𝑔 𝑝 𝑔𝑞 for all 𝑝 and 𝑞); that is
Õ 𝑃 Õ𝑃
𝐻=𝑑 𝑁𝑝 + 𝑔 𝑝 𝑔𝑞 𝐴†𝑝 𝐴𝑞 (5.36)
𝑝=1 𝑝,𝑞=1
However, no exact solution has been discovered for the pairing model with both arbitrary,
𝑝
non-degenerate single-particle orbits 𝑑 𝑝 and arbitrary pairing strengths 𝑔𝑞 (5.2), save for
the case (𝑃 = 2) with only two-energy levels [5]. This lack of an exact solution for the
arbitrary case is what motivates the usage of computational techniques.
80
5.2.2 Full Configuration Interaction
The pairing model can be solved through exact diagonalization. In this method, the
pairing Hamiltonian written in terms of pair fermionic operators (5.2) is represented as a
matrix with elements
𝐻 𝑝𝑖 𝑝 𝑗 ,𝑞𝑖 𝑞 𝑗 = hΦ𝑞𝑖 𝑞 𝑗 |𝐻|Φ 𝑝𝑖 𝑝 𝑗 i
𝑝
= 𝛿 𝑝𝑖 𝑞𝑖 𝛿 𝑝 𝑗 𝑞 𝑗 [2( 𝑝𝑖 + 𝑝 𝑗 ) + 𝑔 𝑝 𝑖𝑗 ]
𝑝 𝑝
+ 𝛿 𝑝𝑖 𝑞𝑖 (1 − 𝛿 𝑝 𝑗 𝑞 𝑗 )𝑔𝑞 𝑗𝑗 + (1 − 𝛿 𝑝𝑖 𝑞𝑖 )𝛿 𝑝 𝑗 𝑞 𝑗 𝑔𝑞𝑖𝑖 , (5.37)
where |Φ 𝑝𝑖 𝑝 𝑗 i = 𝑎 †𝑝𝑖 𝑎 †𝑝 𝑗 |0i. Here, 𝑖 > 𝑗 and |0i is the true vacuum. This Hamiltonian
matrix is then diagonalized, its eigenvalues equal to the possible energies of the system.
5.2.3 Pair CCD
In this section, we apply the method of pair coupled cluster doubles theory (pCCD)
to the pairing model. The pCCD equations (2.161 and 2.162) for the pairing model
Hamiltonian 5.2 become
𝐸 = hΦ0 |𝐻 𝑝 |Φ0 i , (5.38)
0 = hΦ𝑖𝑎 |𝐻 𝑝 |Φ0 i , (5.39)
where the similarity transformed pairing Hamiltonian is
𝐻 𝑝 = 𝑒 −𝑇𝑝 𝐻 𝑝 𝑒𝑇𝑝 , (5.40)
which can be expanded via the BCH identity as
1
𝐻 𝑝 = 𝐻 𝑝 + 𝐻 𝑝 , 𝑇𝑝 + 𝐻 𝑝 , 𝑇𝑝 , 𝑇𝑝 + · · · . (5.41)
2
81
Though this expression is infinite, it can, in this case, be truncated to just the first three
terms. This is because, after the expanded form of the similarity transformed pairing
Hamiltonian (5.41) is inserted into the pCCD equations (5.38 and 5.39), one can truncate
the resulting expressions by noting that only certain terms in the infinite sum (those which
can be fully contracted) are non-zero. The truncated pCCD equations are
𝐸 = hΦ| 𝐹𝑝 + 𝑉𝑝 + 𝑉𝑝 𝑇𝑝 |Φi , (5.42)
1
0 = hΦ𝑖𝑎 | 𝑉𝑝 + 𝐹𝑝 𝑇𝑝 + 𝑉𝑝 𝑇𝑝 + 𝑉𝑝 𝑇𝑝2 − 𝑇𝑝 𝑉𝑝 𝑇𝑝 |Φi , (5.43)
2
where
Õ
𝐹𝑝 = 𝑑𝑝 𝑁𝑝, (5.44)
𝑝
Õ
𝑔𝑞 𝐴†𝑝 𝐴𝑞 ,
𝑝
𝑉𝑝 = (5.45)
𝑝𝑞
with the partitioning of the pairing model Hamiltonian a single-body and two-body term:
𝐻 𝑝 = 𝐹𝑝 + 𝑉𝑝 . We can save ourselves some work by recognizing that the first two terms
of the truncated pCCD energy equation (5.42) equal the reference energy, and hence the
pCCD correlation energy equation is
Δ𝐸 = hΦ|𝑉𝑝 𝑇𝑝 |Φi . (5.46)
Using Wick’s theorem, we calculate the truncated pCCD energy equation (5.42)
Õ
𝑔𝑞 𝑡𝑖𝑎 hΦ| 𝐴†𝑝 𝐴𝑞 𝐴†𝑎 𝐴𝑖 |Φi
𝑝
hΦ|𝑉𝑝 𝑇𝑝 |Φi =
𝑝𝑞𝑖𝑎
Õ
𝑔𝑞 𝑡𝑖𝑎 hΦ|𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑞− 𝑎 𝑞+ 𝑎 †𝑎+ 𝑎 †𝑎− 𝑎𝑖− 𝑎𝑖+ |Φi
𝑝
=
𝑝𝑞𝑖𝑎
82
Õ
𝑝
= 𝑔𝑞 𝑡𝑖𝑎 𝛿 𝑝𝑖 𝛿 𝑞𝑎
𝑝𝑞𝑖𝑎
Õ
= 𝑔𝑖𝑎 𝑡𝑖𝑎 . (5.47)
𝑖𝑎
Turning our attention now to the truncated pCCD amplitude equation (5.42), we start with
Õ
𝑔𝑞 hΦ| 𝐴𝑖† 𝐴𝑎 𝐴†𝑝 𝐴𝑞 |Φi
𝑝
hΦ𝑖𝑎 |𝑉𝑝 |Φi =
𝑝𝑞
Õ
† †
𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑞− 𝑎 𝑞+ |Φi
𝑝
= 𝑔𝑞 hΦ|𝑎𝑖+
𝑝𝑞
Õ
𝑝
= 𝑔𝑞 𝛿 𝑝𝑎 𝛿 𝑞𝑖
𝑝𝑞
= 𝑔𝑖𝑎 . (5.48)
To compute the next term hΦ𝑖𝑎 |𝐹𝑝 𝑇𝑝 |Φi, instead of writing out all possible contractions, it
will be easier to compute the un-truncated term from which the this term comes, namely
hΦ𝑖𝑎 | [𝐹𝑝 , 𝑇𝑝 ] |Φi. We do so by using the commutation relations (5.7 and 5.8) as follows
Õ
hΦ𝑖𝑎 |[𝐹𝑝 , 𝑇𝑝 ] |Φi = 𝑑 𝑝 𝑡 𝑏𝑗 hΦ𝑖𝑎 | [𝑁 𝑝 , 𝐴†𝑏 𝐴 𝑗 ] |Φi
𝑝 𝑗𝑏
Õ
= 𝑑 𝑝 𝑡 𝑏𝑗 hΦ𝑖𝑎 | ( [𝑁 𝑝 , 𝐴†𝑏 ] 𝐴 𝑗 − 𝐴†𝑏 [𝑁 𝑝 , 𝐴 𝑗 ] ) |Φi
𝑝 𝑗𝑏
Õ
=2 𝑑 𝑝 𝑡 𝑏𝑗 (𝛿 𝑝𝑏 − 𝛿 𝑝 𝑗 ) hΦ𝑖𝑎 | 𝐴†𝑏 𝐴 𝑗 |Φi
𝑝 𝑗𝑏
Õ
=2 (𝑑 𝑏 − 𝑑 𝑗 )𝑡 𝑏𝑗 hΦ| 𝐴𝑖† 𝐴𝑎 𝐴†𝑏 𝐴 𝑗 |Φi
𝑗𝑏
Õ
† †
=2 (𝑑 𝑏 − 𝑑 𝑗 )𝑡 𝑏𝑗 hΦ|𝑎𝑖+ 𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑏+ 𝑎 †𝑏− 𝑎 𝑗− 𝑎 𝑗+ |Φi
𝑗𝑏
Õ
=2 (𝑑 𝑏 − 𝑑 𝑗 )𝑡 𝑏𝑗 𝛿𝑖 𝑗 𝛿𝑎𝑏
𝑗𝑏
83
= 2(𝑑 𝑎 − 𝑑𝑖 )𝑡𝑖𝑎 . (5.49)
The same is true for the next term hΦ𝑖𝑎 |𝑉𝑝 𝑇𝑝 |Φi and so it will be calculated analogously
Õ
𝑔𝑞 𝑡 𝑏𝑗 hΦ𝑖𝑎 |[ 𝐴†𝑝 𝐴𝑞 , 𝐴†𝑏 𝐴 𝑗 ] |Φi
𝑝
hΦ𝑖𝑎 | [𝑉𝑝 , 𝑇𝑝 ] |Φi =
𝑝𝑞 𝑗 𝑏
Õ
𝑔𝑞 𝑡 𝑏𝑗 hΦ𝑖𝑎 | ( 𝐴†𝑝 𝐴†𝑏 ] + 𝐴†𝑝 [ 𝐴𝑞 , 𝐴†𝑏 ] 𝐴 𝑗
𝑝
= [ 𝐴𝑞 ,𝐴𝑗
𝑝𝑞 𝑗 𝑏
+ 𝐴†𝑏 [ 𝐴†𝑝 , 𝐴 𝑗 ] 𝐴𝑞 + [ 𝐴†𝑝 †
]
𝑏 𝐴 𝑗 𝐴𝑞 )
|Φi
, 𝐴
Õ
𝑔𝑞 𝑡 𝑏𝑗 hΦ𝑖𝑎 | (𝛿 𝑞𝑏 𝐴†𝑝 (1 − 𝑁 𝑏 ) 𝐴 𝑗 + 𝛿 𝑝 𝑗 𝐴†𝑏 (𝑁 𝑗 − 1) 𝐴𝑞 ) |Φi
𝑝
=
𝑝𝑞 𝑗 𝑏
Õ
𝑔𝑞 𝑡 𝑏𝑗 hΦ𝑖𝑎 | (𝛿 𝑞𝑏 𝐴†𝑝 (1 − 𝑁 𝑏 ) 𝐴 𝑗 + 𝛿 𝑝 𝑗 𝐴†𝑏 𝐴𝑞 (𝑁 𝑗 − 1) − 2𝛿 𝑝 𝑗 𝛿 𝑞 𝑗 𝐴†𝑏 𝐴 𝑗 ) |Φi
𝑝
=
𝑝𝑞 𝑗 𝑏
Õ
𝑡 𝑏𝑗 hΦ| 𝐴𝑖† 𝐴𝑎 (𝑔𝑏 𝐴†𝑝 𝐴 𝑗 + 𝑔𝑞 𝐴†𝑏 𝐴𝑞 − 2𝑔 𝑗 𝐴†𝑏 𝐴 𝑗 ) |Φi
𝑝 𝑗 𝑗
=
𝑗𝑏
Õ
† †
𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑗− 𝑎 𝑗+ |Φi
𝑝
= 𝑡 𝑏𝑗 (𝑔𝑏 hΦ|𝑎𝑖+
𝑗𝑏
† †
𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑏+ 𝑎 †𝑏− 𝑎 𝑞− 𝑎 𝑞+ )|Φi
𝑗
+ 𝑔𝑞 hΦ|𝑎𝑖+
† †
𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑏+ 𝑎 †𝑏− 𝑎 𝑗− 𝑎 𝑗+ )|Φi
𝑗
− 2𝑔 𝑗 hΦ|𝑎𝑖+
Õ
𝑝 𝑗 𝑗
= 𝑡 𝑏𝑗 (𝑔𝑏 𝛿𝑖 𝑗 𝛿𝑎 𝑝 + 𝑔𝑞 𝛿𝑖𝑞 𝛿𝑎𝑏 − 2𝑔 𝑗 𝛿𝑖 𝑗 𝛿𝑎𝑏 )
𝑗𝑏
Õ Õ
𝑗
= 𝑔𝑏𝑎 𝑡𝑖𝑏 + 𝑔𝑖 𝑡 𝑎𝑗 − 2𝑔𝑖𝑖 𝑡𝑖𝑎 , (5.50)
𝑏 𝑗
where we’ve used the facts that [𝑁 𝑏 , 𝐴 𝑗 ] = 0, [𝑁 𝑗 , 𝐴𝑞 ] = −2𝛿 𝑞 𝑗 𝐴 𝑗 , and (1 − 𝑁 𝑏 ) |Φi =
(𝑁 𝑗 − 1) |Φi = 1. We will skip the penultimate term for now, the reasons for which will
84
become clear later, and compute the final term first
− Φ𝑖𝑎 𝑇𝑝 𝑉𝑝 𝑇𝑝 Φ (5.51)
Õ
𝑔𝑞 𝑡 𝑏𝑗 𝑡 𝑘𝑐 hΦ| 𝐴𝑖† 𝐴𝑎 𝐴†𝑏 𝐴 𝑗 𝐴†𝑝 𝐴𝑞 𝐴†𝑐 𝐴 𝑘 |Φi
𝑝
= −
𝑝𝑞 𝑗 𝑘 𝑏𝑐
Õ
𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑏+ 𝑎 †𝑏− 𝑎 𝑗− 𝑎 𝑗+ 𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑞− 𝑎 𝑞+ 𝑎 †𝑐+ 𝑎 †𝑐− 𝑎 𝑘− 𝑎 𝑘+ |Φi
† †
𝑝
= − 𝑔𝑞 𝑡 𝑏𝑗 𝑡 𝑘𝑐 hΦ|𝑎𝑖+
𝑝𝑞 𝑗 𝑘 𝑏𝑐
Õ
𝑝
= − 𝑔𝑞 𝑡 𝑏𝑗 𝑡 𝑘𝑐 𝛿𝑖 𝑗 𝛿𝑎𝑏 𝛿 𝑝𝑘 𝛿 𝑞𝑐
𝑝𝑞 𝑗 𝑘 𝑏𝑐
Õ
= − 𝑔𝑐𝑘 𝑡𝑖𝑎 𝑡 𝑘𝑐
𝑘𝑐
Õ
𝑗
= − 𝑔𝑏 𝑡𝑖𝑎 𝑡 𝑏𝑗 , (5.52)
𝑗𝑏
where we’ve relabeled 𝑘 → 𝑗 and 𝑐 → 𝑏. We now come to the final term hΦ𝑖𝑎 |𝑉𝑝 𝑇𝑝2 |Φi,
which we left for last as it is most easily solved through diagrammatic methods due to the
large number of possible contractions. The term is calculated up to the point of contractions
as
1 𝑎
hΦ | 𝑉𝑝 𝑇𝑝2 |Φi
2 𝑖
1 Õ 𝑝 𝑏 𝑐
= 𝑔𝑞 𝑡 𝑗 𝑡 𝑘 hΦ| 𝐴𝑖† 𝐴𝑎 𝐴†𝑝 𝐴𝑞 𝐴†𝑏 𝐴 𝑗 𝐴†𝑐 𝐴 𝑘 |Φi
2 𝑝𝑞 𝑗 𝑘 𝑏𝑐
1 Õ 𝑝 𝑏 𝑐 † †
= 𝑔 𝑡 𝑡 hΦ| 𝑎𝑖+ 𝑎𝑖− 𝑎 𝑎− 𝑎 𝑎+ 𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑞− 𝑎 𝑞+ 𝑎 †𝑏+ 𝑎 †𝑏− 𝑎 𝑗− 𝑎 𝑗+ 𝑎 †𝑐+ 𝑎 †𝑐− 𝑎 𝑘− 𝑎 𝑘+ |Φi ,
2 𝑝𝑞 𝑗 𝑘 𝑏𝑐 𝑞 𝑗 𝑘
(5.53)
at which point we use a novel extension of Goldstone diagrams (which I’ve named pair-
Goldstone diagrams) to continue.
In standard Goldstone diagrams, vertices represent macro-operators (like 𝐹𝑝 , 𝑉𝑝 , and
𝑇 as defined above) while lines between vertices represent contractions between fermionic
85
operators contained within the macro-operators represented by said vertices. Lines directed
to the right or left represent contractions between hole and particle operators, respectively.
The resulting expressions can be read directly from the diagrams by identifying lines
entering or leaving a vertex as the lower or upper index of the prefactor represented by said
vertex, respectively.
However, standard Goldstone diagrams do not visually capture additional Kronecker
delta’s that can be created when dealing with pairing Hamiltonians. To demonstrate,
consider Figure 5.2a which is a traditional Goldstone diagram that represents one possible
set of contractions that result from (5.53) which comes from the term hΦ𝑖𝑎 |𝑉𝑝 𝑇𝑝2 |Φi. In this
diagram, the top left vertex represents the excitation operator 𝐸 𝑎𝑖 = 𝐴𝑖† 𝐴𝑎 (which creates
the excited state hΦ| 𝐸 𝑎𝑖 = hΦ𝑖𝑎 |), the bottom left vertex represents 𝑉𝑝 , and the top and
bottom right diagrams represent the two 𝑇𝑝 diagrams. This particular diagram represents
the contractions that result in the product
𝛿𝑖+ 𝑗+ 𝛿𝑎+𝑏+ 𝛿𝑎−𝑏− 𝛿𝑖−𝑘− 𝛿 𝑝− 𝑗− 𝛿 𝑞−𝑐− 𝛿 𝑞+𝑐+ 𝛿 𝑝+𝑘+ , (5.54)
which results in the following transformation
Õ Õ
𝑝
𝑔𝑞 𝑡 𝑏𝑗 𝑡 𝑘𝑐 → 𝑔𝑖𝑐 𝑡𝑖𝑎 𝑡𝑖𝑐 , (5.55)
𝑝𝑞 𝑗 𝑘 𝑏𝑐 𝑗𝑐
which is equivalent, upon relabeling 𝑐 → 𝑏, to
Õ
𝑔𝑖𝑏 𝑡𝑖𝑎 𝑡𝑖𝑏 , (5.56)
𝑗𝑏
which is immediately read from the diagram when labeling the rightward facing arrows
(from top to bottom) 𝑖+, 𝑖−, 𝑗−, and 𝑗+ and the leftward facing arrows (from top to bottom)
𝑎+, 𝑎−, 𝑏−, and 𝑏+.
86
Figure 5.2: Pair-Goldstone contraction schematic.
Note that we actually reduced the number of summed indices by three despite only
having eight contractions (which would normally result in a reduction of 2 indices). This
occurs because we are working with a pair operators. To see this, note that the products
𝛿𝑖+ 𝑗+ 𝛿𝑖−𝑘− and 𝛿 𝑝− 𝑗− 𝛿 𝑝+𝑘+ imply 𝛿 𝑝𝑖 𝛿 𝑗𝑖 𝛿 𝑘𝑖 . That is, the indices 𝑝, 𝑗, and 𝑘, all go to 𝑖. This
"additional" contraction is not immediately obvious from the traditional Goldstone diagram
(Figure 5.2a). However, it is in the pair-Goldstone diagram representation (Figures 5.2b
and 5.2c). Here, each vertex is replaced with a solid line, one end of which is exclusively
for particle lines while the other is exclusively for hole lines. There will always be two
lines attached to each end; one for the + state of the pair and the other for the − state. This
allows one to visually capture the phenomenon of pairs of fermionic operators contracting
to operators with differing indices. Notice that in Figure 5.2b, the hole lines of the top 𝑇𝑝
operator are 𝑖+ and 𝑗−, while the hole lines of the bottom 𝑇𝑝 operator are 𝑖− and 𝑗+. This
clearly implies that the bottom indices of both 𝑡 terms should be 𝑖. This is visually shown
in the "contracted diagram" Figure 5.2c, in which the hole ends of the 𝑇𝑝 lines have been
brought or "contracted" together. This additionally results in the merging of the 𝑖+ and
𝑗− lines, as well as the 𝑖− and 𝑗+ lines. The lines are now labeled (from top to bottom)
87
𝑖+, 𝑎+, 𝑎−, 𝑏−, 𝑏+, 𝑖−. Note that because of all this, the final expression (5.56) can be
immediately read from the pair-Goldstone representation (Figure 5.2c).
We now use pair-Goldstone diagrams to calculate the term hΦ𝑖𝑎 | 𝑉𝑝 𝑇𝑝2 |Φi (5.53). That
is, the pair-Goldstone diagrams from Figure 5.3 result in the following expressions
Õ
𝑗
𝑎) → 2 𝑔𝑏 𝑡 𝑏𝑗 𝑡𝑖𝑎 (5.57)
𝑏𝑗
Õ
𝑗
𝑏) → 2 𝑔𝑏 𝑡 𝑎𝑗 𝑡𝑖𝑏 (5.58)
𝑏𝑗
Õ
𝑐) → −4 𝑔𝑖𝑏 𝑡𝑖𝑏 𝑡𝑖𝑎 (5.59)
𝑏
Õ
𝑗
𝑑) → −4 𝑔𝑎 𝑡 𝑎𝑗 𝑡𝑖𝑎 (5.60)
𝑗
𝑒) → 4𝑔𝑖𝑎 𝑡𝑖𝑎 𝑡𝑖𝑎 . (5.61)
The minus signs in front of the expressions resulting from diagrams c) and d) come from
the fact that their pre-contracted diagrams (see for example Figure 5.2b) contain an odd
number of crossings. This can be immediately obtained from the contracted diagrams by
counting the number of pairs of ends of lines that have been merged; in diagrams c) and d),
one pair of ends has been merged (either the hole ends or the particle ends) resulting in a
prefactor sign of (−1) 1 , while in diagram e), two pairs of ends have been merged (both the
hole ends and the particle ends) resulting in a prefactor sign of (−1) 2 . The pre-factors of the
expressions resulting from diagrams a) and b) can be determined by counting the number
of symmetries in each diagram (the same as in standard Goldstone diagrams) for which
each is 2. The pre-factors of the contracted diagrams c), d), and e) can be determined
by counting the number of unique ways that each of these diagrams can be created by
contracting the un-contracted diagrams, a) and b), for which each is four (two from each
88
Figure 5.3: Pair-Goldstone diagrams for the pairing model.
un-contracted diagram).
Plugging these resulting expression (5.57)-(5.61) into equation (5.53) gives us
1 𝑎 Õ
𝑗
Õ
𝑗
Õ Õ
𝑗
hΦ𝑖 | 𝑉𝑝 𝑇𝑝2 |Φi = 𝑔𝑏 𝑡 𝑏𝑗 𝑡𝑖𝑎 + 𝑔𝑏 𝑡 𝑎𝑗 𝑡𝑖𝑏 − 2 𝑔𝑖𝑏 𝑡𝑖𝑏 𝑡𝑖𝑎 − 2 𝑔𝑎 𝑡 𝑎𝑗 𝑡𝑖𝑎 + 2𝑔𝑖𝑎 𝑡𝑖𝑎 𝑡𝑖𝑎 .
2 𝑏𝑗 𝑏𝑗 𝑏 𝑗
(5.62)
All together, the pCCD correlation energy equation (5.46) becomes
Õ
Δ𝐸 = 𝑔𝑖𝑎 𝑡𝑖𝑎 , (5.63)
𝑖𝑎
while the pCCD amplitude equations (5.43) become
Õ Õ Õ Õ Õ
𝑗 𝑗 𝑗
0 = 𝑔𝑖𝑎 + 2 𝑑 𝑎 − 𝑑𝑖 − 𝑔𝑖𝑖 + 𝑔𝑖𝑎 𝑡𝑖𝑎 − 𝑔𝑖𝑏 𝑡𝑖𝑏 − 𝑔𝑎 𝑡 𝑎𝑗 𝑡𝑖𝑎 + 𝑔𝑏𝑎 𝑡𝑖𝑏 + 𝑔𝑖 𝑡 𝑎𝑗 + 𝑔𝑏 𝑡 𝑎𝑗 𝑡𝑖𝑏 .
𝑏 𝑗 𝑏 𝑗 𝑏𝑗
(5.64)
89
In practice, the amplitudes 𝑡 are solved for by solving the amplitude equation (5.64), which
is non-linear and coupled, via an iterative root finding algorithm such as Newton’s method
and plugging them into the energy equation (5.63).
Although we were able to skip over deriving the reference energy of the pairing model
in the previous section because we only cared about the correlation energy Δ𝐸, we will need
an expression for the reference energy when we turn to implementing unitary pair coupled
cluster (UpCC) theory as the ansatz for the variational quantum eigensolver (VQE). Thus,
we derive said expression here using Wick’s theorem as here
𝐸 ref = hΦ|𝐻 𝑝 |Φi
= hΦ|𝐹𝑝 |Φi + hΦ|𝑉𝑝 |Φi
Õ Õ
𝑑 𝑝 hΦ|𝑎 †𝑝𝜎 𝑎 𝑝𝜎 |Φi + 𝑔𝑞 hΦ|𝑎 †𝑝+ 𝑎 †𝑝− 𝑎 𝑞− 𝑎 𝑞+ |Φi
𝑝
=
𝑝𝜎 𝑝𝑞
Õ Õ
𝑝
=2 𝑑 𝑝 ℎ( 𝑝) + 𝑔𝑞 ℎ( 𝑝)𝛿 𝑝𝑞
𝑝 𝑝𝑞
Õ
= (2𝑑𝑖 + 𝑔𝑖𝑖 ). (5.65)
𝑖
In order to find the amplitudes 𝑡, Newton’s method must converge, which requires a good
initial guess for each 𝑡. To find one, we turn to many-body perturbation theory.
5.2.4 Many-body Perturbation Theory
The single particle energies 𝜖 𝑝 (2.107) for the pairing Hamiltonian are given by
Õ
𝑝 𝑝𝑖
𝜖𝑝 = 𝑡𝑝 + 𝑣 𝑝𝑖 (5.66)
𝑖
Õ
𝑝
= 𝑑𝑝 + 𝛿 𝑝𝑖 𝑔 𝑝 (5.67)
𝑖
𝑝
= 𝑑 𝑝 + ℎ( 𝑝)𝑔 𝑝 , (5.68)
90
and the interacting Hamiltonian 𝐻 𝐼 is given by
Õ 𝑃 Õ𝑃
𝑔𝑞 𝐴†𝑝 𝐴𝑞 .
𝑝
𝐻𝐼 = (𝑑 𝑝 − 𝜖 𝑝 )𝑁 𝑝 + (5.69)
𝑝=1 𝑝,𝑞=1
For the pairing model, the second and third order energy correlation contributions (2.155
- 2.156) are
Õ 𝑔 𝑎 𝑔𝑖𝑎
Δ𝐸 (2) = 𝑖
𝑎𝑖
𝜖𝑖𝑎
1Õ 𝑔𝑖𝑎 𝑔𝑖𝑎
= , (5.70)
2 𝑎𝑖 𝑑𝑖 − 𝑑 𝑎 + 𝑔𝑖𝑖
and
Õ 𝑔𝑖 𝑔𝑎𝑏 𝑔 𝑎 Õ 𝑔 𝑎𝑗 𝑔𝑖𝑗 𝑔𝑖𝑎
(3) 𝑏 𝑖
Δ𝐸 = +
𝑖𝑎𝑏
𝜖𝑖𝑎 𝜖𝑖𝑏 𝑖 𝑗𝑎
𝜖𝑖𝑎 𝜖 𝑎𝑗
𝑗
1Õ 𝑔𝑖𝑏 𝑔𝑎𝑏 𝑔𝑖𝑎 Õ 𝑔 𝑎𝑗 𝑔𝑖 𝑔𝑖𝑎
= + . (5.71)
4 𝑖𝑎𝑏 𝑑𝑖 − 𝑑 𝑎 + 𝑔𝑖𝑖 𝑑𝑖 − 𝑑 𝑏 + 𝑔𝑖𝑖 𝑖 𝑗 𝑎 𝑑𝑖 − 𝑑 𝑎 + 𝑔𝑖
𝑖 𝑗
𝑑 𝑗 − 𝑑𝑎 + 𝑔 𝑗
One good initial guess for 𝑡 can be found by mathcing the pCCD correlation energy (5.63)
with the second order contribution to the correlation energy from MBPT (5.70)
Õ Õ 𝑔𝑖𝑎
𝑔𝑖𝑎 𝑡𝑖𝑎 (0) = 𝑔𝑖𝑎 , (5.72)
𝑖𝑎 𝑖𝑎
𝜖𝑖𝑎
which allows us to identify
𝑔𝑖𝑎
𝑡𝑖𝑎 (0) =
𝜖𝑖𝑎
1 𝑔𝑖𝑎
= , (5.73)
2 𝑑𝑖 − 𝑑 𝑎 + 𝑔𝑖𝑖
for the pairing model. Many-body perturbation theory provides a more tractable, yet still
approximate, solution.
91
5.3 Quantum Solutions
5.3.1 Mapping the Hamiltonian
To solve the pairing model on a quantum computer, we must first determine a mapping
of our problem to qubits. To do so, we simply let qubit 𝑝 represent energy level 𝑝. In our
pairing scheme, the qubit being in state |0i or |1i implies that its corresponding energy
level is completely unoccupied or occupied by a pair of fermions, respectively. Note that
there is no possibility for broken pairs in this mapping. Now that we’ve chosen a qubit
mapping, we must transform the Hamiltonian accordingly, from fermionic operators to spin
operators. To do so, we’ll first separate the pairing Hamiltonian (5.2) into strict one-body
and two-body terms, as follows
𝑃 𝑃
1Õ Õ
𝑔𝑞 𝐴†𝑝 𝐴𝑞 ,
𝑝 𝑝
𝐻= (2𝑑 𝑝 + 𝑔 𝑝 )𝑁 𝑝 + (5.74)
2 𝑝=1 𝑝,𝑞=1
𝑝≠𝑞
which follows from the fact that, when 𝑝 = 𝑞 in the second sum of (5.2), it results in the
operator
𝑋 𝑝 − 𝑖𝑌𝑝 𝑋 𝑝 + 𝑖𝑌𝑝
𝐴†𝑝 𝐴 𝑝 = (5.75)
2 2
𝐼𝑝 − 𝑍𝑝
= (5.76)
2
𝑁𝑝
= . (5.77)
2
Using the pair-fermionic anti-commutation relation (5.6) and applying the Jordan-Wigner
transformation to (5.77) yields
𝑁𝑝 1
= 𝐼 𝑝 − 𝛿 𝑝𝑞 [ 𝐴 𝑝 , 𝐴†𝑞 ] (5.78)
2 2
1 1
= 𝐼 𝑝 − 𝛿 𝑝𝑞 [𝑋 𝑝 + 𝑖𝑌𝑝 , 𝑋𝑞 − 𝑖𝑌𝑞 ] (5.79)
2 4
92
𝐼𝑝 − 𝑍𝑝
= . (5.80)
2
To deal with the second term of (5.74), we note that the sum can be broken up as
Õ 𝑃 Õ 𝑃 Õ𝑃
𝑔𝑞 𝐴†𝑝 𝐴𝑞 = 𝑔𝑞 𝐴†𝑝 𝐴𝑞 + 𝑔𝑞 𝐴†𝑝 𝐴𝑞
𝑝 𝑝 𝑝
(5.81)
𝑝,𝑞=1 𝑝,𝑞=1 𝑝,𝑞=1
𝑝≠𝑞 𝑝<𝑞 𝑞<𝑝
Õ 𝑃 Õ𝑃
𝑔𝑞 𝐴†𝑝 𝐴𝑞 𝑔𝑞 𝐴†𝑞 𝐴 𝑝
𝑝 𝑝
= + (5.82)
𝑝,𝑞=1 𝑝,𝑞=1
𝑝<𝑞 𝑝<𝑞
Õ 𝑃
𝑔𝑞 ( 𝐴†𝑝 𝐴𝑞 + 𝐴 𝑝 𝐴†𝑞 ),
𝑝
= (5.83)
𝑝,𝑞=1
𝑝<𝑞
where we’ve swapped the indices 𝑝 ↔ 𝑞 to obtain (5.82) and used the fact that [ 𝐴 𝑝 , 𝐴†𝑞 ] =
0 for 𝑝 ≠ 𝑞 (which 𝑝 < 𝑞 implies) to obtain (5.83). Applying the Jordan-Wigner
transformation to (5.83) yields
𝑋 𝑝 − 𝑖𝑌𝑝 𝑋𝑞 + 𝑖𝑌𝑞 𝑋 𝑝 + 𝑖𝑌𝑝 𝑋𝑞 − 𝑖𝑌𝑞
𝐴†𝑝 𝐴𝑞 + 𝐴 𝑝 𝐴†𝑞 = + (5.84)
2 2 2 2
𝑋 𝑝 𝑋𝑞 + 𝑌𝑝𝑌𝑞
= . (5.85)
2
All together, the pairing Hamiltonian (5.74), after Jordan-Wigner transformation, becomes
𝑃
1Õ 𝑝 1 Õ 𝑝
𝐻= (2𝑑 𝑝 + 𝑔 𝑝 )(𝐼 𝑝 − 𝑍 𝑝 ) + 𝑔 (𝑋 𝑝 𝑋𝑞 + 𝑌𝑝𝑌𝑞 ). (5.86)
2 𝑝=1 2 𝑝,𝑞=1 𝑞
𝑝<𝑞
5.3.2 Mapping the Ansatz
As an ansatz for the pairing model, we choose the unitary pair coupled cluster doubles
(UpCCD) ansatz, which is the unitary version of the pair coupled cluster doubles (pCCD)
ansatz. It is chosen because any quantum ansatz must be unitary (as all quantum gates
93
must be unitary), it is a pairing ansatz, and includes only up to doubles which allows for
the use of, at most, efficiently decomposable two-qubit gates. It is given by
†
|Ψi = 𝑒𝑇𝑝 −𝑇𝑝 |Φi . (5.87)
Recalling that 𝑇𝑝 , as defined in (2.168), is given by
Õ
𝑇𝑝 = 𝑡𝑖𝑎 𝐴†𝑎 𝐴𝑖 , (5.88)
𝑖𝑎
turns the ansatz into the variational form
( )
Õ
|Ψ(𝑡)i = exp 𝑡𝑖𝑎 𝐴†𝑎 𝐴𝑖 − 𝐴𝑎 𝐴𝑖† |Φ0 i , (5.89)
𝑖𝑎
where we’ve used the fact that [ 𝐴𝑎 , 𝐴𝑖† ] = 0 for since, by definition 𝑎 ≠ 𝑖. Applying the
Jordan Wigner transformation to the operators yields
† † 𝑋𝑎 − 𝑖𝑌𝑎 𝑋𝑖 + 𝑖𝑌𝑖 𝑋𝑎 + 𝑖𝑌𝑎 𝑋𝑖 − 𝑖𝑌𝑖
𝐴𝑎 𝐴𝑖 − 𝐴𝑎 𝐴𝑖 = − (5.90)
2 2 2 2
𝑖
= (𝑋𝑎𝑌𝑖 − 𝑌𝑎 𝑋𝑖 ). (5.91)
2
Plugging this back into the ansatz (5.89) yields
( )
𝑖 Õ 𝑎
|Ψ(𝑡)i = exp 𝑡 (𝑋𝑎𝑌𝑖 − 𝑌𝑎 𝑋𝑖 ) |Φ0 i . (5.92)
2 𝑖𝑎 𝑖
As it is not known how to efficiently decompose n-qubit operators where 𝑛 is greater than
two, we must write our ansatz as a sum of two-qubit operators. To achieve this, we employ
a simple Suzuki-Trotter approximation [82], which we truncate to a single first-order step
in order to minimize the depth of the quantum circuit that will implement it. Doing so
yields
Ö
|Ψ(𝑡)i = 𝐴𝑖𝑎 |Φ0 i , (5.93)
𝑖𝑎
94
where we’ve defined the two-qubit operator
𝑎 𝑖 𝑎
𝐴𝑖 = exp 𝑡𝑖 (𝑋𝑎𝑌𝑖 − 𝑌𝑎 𝑋𝑖 ) , (5.94)
2
which is efficiently decomposable. One could consider higher order Trotterizations or
multiple Trotter steps; however, employing these would substantially increase the depth
of the circuit. Additionally, it will be seen later that our simple Trotter approximation
provides a sufficient ansatz for VQE to be able to minimize the energy. The operator 𝐴𝑖𝑎
takes the following matrix form
©1 0 0 0ª
®
0 cos 𝑡 𝑎𝑖 sin 𝑡 𝑎𝑖
®
𝑎
0®
𝐴𝑖 = ®.
® (5.95)
0 − sin 𝑡 𝑎𝑖 cos 𝑡 𝑎𝑖 0®®
®
«0 0 0 1¬
From this matrix form, one can see that 𝐴𝑖𝑎 acts non-trivially only in the two-qubit subspace
{|01i , |10i} and is closed in said subspace. This means that as long as we initialize our
quantum circuit to a bit-string state with Hamming weight equal to the number of particles,
we’re guaranteed (save for noise) to only search the relevant subspace of the Hamiltonian.
5.3.3 Implementing the Ansatz
Now that we’ve mapped the ansatz to spin operators, we must now determine how to
efficiently decompose it into a quantum circuit given the limitations of qubit connectivity
and circuit depth allowed on NISQ era devices. Here, we give two such methods; the first
is for a quantum computer with linear connectivity while the second is for one with circular
connectivity.
To define these two connectivity terms, consider a quantum computer with 𝑁 qubits.
Label and order the qubits as 0, 2, ..., 𝑁 − 1. A quantum computer has linear connectivity
95
𝑞1 𝑞2 ··· 𝑞𝑁
Figure 5.4: Linear qubit-connectivity schematic.
𝑞1
𝑞𝑁 𝑞2
···
Figure 5.5: Circular qubit-connectivity schematic.
when each of its qubits, except for the first and last, is connected to its left and right
neighboring qubits. The first and last qubits are connected to their only neighboring qubit.
This is visualized in Figure 5.4 where the qubits are connected if the circles representing
them are connected in the graph. A quantum computer has circular connectivity when
each of its qubits (including the first and last) is connected to its left and right neighboring
qubits. This is visualized in Figure 5.5. The ansatz is implemented using what we’ll call
the particle-hole swap network (phsn) technique which has both a linear connectivity [30]
and a novel circular connectivity version. An illustration of the circuits for the four-particle,
five-hole system for both connectivities are given in Figure 5.6.
In said circuits, 𝑆 is the SWAP gate 3.37 and 𝐴𝑖𝑎 is the two qubit operator defined
in (5.94). Because 𝑆 𝐴𝑖𝑎 ∈ SO(4), it can be efficiently mapped to a depth five circuit
consisting of two CNOTs and twelve single-qubit gates [83]. With 𝑃 being the number of
energy levels and 𝑁 being the number of energy levels that are initially filled, the number
of two-qubit gates 𝑆 𝐴𝑖𝑎 is equal to 𝐺 = 𝑃(𝑃 − 𝑁). The depth of the linear particle-hole
swap network is 𝐷 (lphsn) = 𝑃 − 1 where 𝑃 is the number of qubits (energy-levels in
the pairing model). The depth of the circular particle-hole swap network, however, is
96
(a) (b)
Figure 5.6: a) Linear particle-hole swap network (lphsn) for a four-particle, five-hole
system. b) Circular particle-hole swap network for a four-particle, five-hole system (cphsn).
See Figure 5.7 for schematic representation.
𝐷 (cphsn) = max(𝑁, 𝑃 − 𝑁) where 𝑁 is the number of particles (Hamming weight of the
initial state). Because 𝑁 is bounded as 1 ≤ 𝑁 ≤ 𝑃, we have that the depth of the cphsn is
bounded as d𝑃/2e ≤ 𝐷 (cphsn) ≤ 𝑃 − 1. The depth decreases as 𝑁 approached 𝑃/2 from
either direction, achieving a minimum of d𝑃/2e. Thus, we have that 𝐷 (cphsn) ≤ 𝐷 (lphsn).
That is, the circular connectivity enables a circuit with depth less than or equal to that of
a circuit constrained by linear connectivity, up to reduction by a factor of 2. Shortening
the depth of the circuit is vital for decreasing noise on NISQ era devices. The algorithm to
implement the linear particle-hole swap network is given below.
5.3.4 Initialization
As VQE involves a minimization algorithm, it is important that we choose a good initial
guess for the variational parameters 𝑡𝑖𝑎 so as to give the classical minimization algorithm
the best chance of finding the ground state energy quickly and with high precision. The
97
(a) (b)
Figure 5.7: a) Schematic representation of the linear particle-hole swap network (lphsn) for
a four-particle, five-hole system. b) Schematic representation of the circular particle-hole
swap network (cphsn) for a four-particle, five-hole system. Each circle represents a qubit
and a slot (particle/hole). The particles are labeled 𝑝 0 , ..., 𝑝 3 and are colored blue while
the holes are labeled ℎ0 , ..., ℎ4 and are colored red. The first and last columns of circles
are the initial and final positions of the qubits/slots. A rectangle around a pair of circles
𝑝
( 𝑝𝑖 , ℎ 𝑗 ) denotes that the gate 𝑆 𝐴 ℎ 𝑖𝑗 has been applied between the corresponding qubits. See
Figure 5.6 for circuit representation.
initial guess we will be using is
1 𝑔𝑖𝑎
𝑡𝑖𝑎 (0) = , (5.96)
2 𝑑𝑖 − 𝑑 𝑎 + 𝑔𝑖𝑖
from (5.73) which was computed by comparing the pCCD correlation energy with the
second order contribution to the correlation energy from MBPT in subsection 5.2.4. This
provides a good initial guess because we know that it results in an energy that is close
to the second order contribution to the correlation energy from MBPT, which in turn is
a good approximation to the true ground state. This means that our minimizer will start
close to the correct solution in the energy landscape through which it must transverse. It
also implies that, barring degeneracy, an ansatz initialized to 𝑡𝑖𝑎 (0) should have a significant
overlap with the true ground state wavefunction.
98
Algorithm 5.1 Linear Particle-Hole Swap Network
Input: Number of energy levels 𝑝, number of initially filled energy levels 𝑛,
gates SA𝑖𝑎 , and qubit order list
𝑜 = [1, 2, ..., 𝑛].
Output: Quantum circuit implementation of lphsn.
𝑡 =𝑛−1 ⊲ index of highest qubit acted upon
𝑏=𝑛 ⊲ index of lowest qubit acted upon
ℎ=1 ⊲ height (# of gates per column)
𝑑𝑡 = −1 ⊲ top direction
𝑑𝑏 = 1 ⊲ bottom direction
for 0 ≤ 𝑖 ≤ 𝑛 − 1 do
apply gate 𝑋 to qubit 𝑖
end for
while ℎ ≠ 0 do
for 0 ≤ 𝑖 ≤ ℎ − 1 do
𝑥 = 𝑡 + 2𝑖
apply SA𝑜[𝑥]
𝑜[𝑥+1]
to qubits 𝑥 and 𝑥 + 1
𝑜[𝑥], 𝑜[𝑥 + 1] = 𝑜[𝑥 + 1], 𝑜[𝑥]
end for
if 𝑡 = 1 then
𝑑𝑡 = 1
end if
if 𝑏 = 𝑝 then
𝑑 𝑏 = −1
end if
𝑡 += 𝑑𝑡 , 𝑏 += 𝑏 𝑑 , ℎ = (𝑏 − 𝑡 + 1)/2
end while
Figure 5.8 compares initial correlation energies h𝜓(𝜃)|𝐻|𝜓(𝜃)i between the case where
the initial parameters 𝑡𝑖𝑎 are chosen randomly and where they are informed by many-body
perturbation theory, 𝑡𝑖𝑎 = 𝑡𝑖𝑎 (0) (5.96); It uses the pairing model with 𝑃 = 4 energy levels
and 𝑁 = 2 pairs of particles. It can be seen from said figure that the initial correlation
energy resulting from 𝑡𝑖𝑎 = 𝑡𝑖𝑎 (0) (E_calc_ia) is much closer to the correct ground state
correlation energy (E_true) than the initial correlation energy resulting from random initial
99
Algorithm 5.2 Circular Particle-Hole Swap Network
Input: Number of energy levels 𝑝, number of initially
filled energy levels 𝑛, gates SA𝑖𝑎 , and qubit order list
𝑜 = [1, 2, ..., 𝑛].
Output: Quantum circuit implementation of cphsn.
𝑑 = max(𝑛, 𝑝 − 𝑛) ⊲ depth of circuit
ℎ = min(𝑛, 𝑝 − 𝑛) ⊲ height (# of gates per column)
for 0 ≤ 𝑖 ≤ 𝑛 − 1 do
apply gate 𝑋 to qubit 2𝑖
end for
if 𝑛 ≥ d𝑝/2e then
for 0 ≤ 𝑖 ≤ 𝑝 − 1 do
apply gate 𝑋 to qubit 𝑖
end for
end if
for 0 ≤ 𝑖 ≤ 𝑑 − 1 do
for 0 ≤ 𝑖 ≤ ℎ − 1 do
𝑥 = (𝑖 + 2 𝑗) mod 𝑝
apply SA𝑜[𝑥]
𝑜[𝑥+1]
to qubits 𝑥 and 𝑥 + 1
𝑜[𝑥], 𝑜[𝑥 + 1] = 𝑜[𝑥 + 1], 𝑜[𝑥]
end for
end for
parameters (E_calc_rand). In fact, the randomness of the initial parameters case seems to
carry over to the corresponding initial correlation energy as it produces a noisy line that
is way above the correct correlation energies. Here, we’ve defined the initial correlation
energies as
D E
𝐸 calc_ia = Φ 𝑡𝑖𝑎 (0) 𝐻Φ 𝑡𝑖𝑎 (0) , (5.97)
𝐸 calc_rand = hΦ (𝑡rand )|𝐻|Φ (𝑡 rand )i , (5.98)
𝐸 true = hΦ (𝑡min )|𝐻|Φ (𝑡min )i , (5.99)
where 𝑡rand is a set of random parameters while 𝑡min is the set of parameters that minimize
the energy.
100
Figure 5.8: Initial correlation energies for the pairing model with 𝑃 = 4 energy levels
and 𝑁 = 2 pairs of fermions are compared to E_true, the true ground state correlation
energy. E_calc_ia uses the initial parameters informed by MBPT (5.96) and E_calc_rand
uses random initial parameters.
Because (E_calc_ia) is so much closer to (E_true) than (E_calc_rand), the optimization
portion of the algorithm should not have to go through as many iterations when using
𝑡𝑖𝑎 = 𝑡𝑖𝑎 (0) as opposed to 𝑡𝑖𝑎 = 𝑡rand which is confirmed by Figure 5.9. This illustrates
the importance of initializing one’s variational parameters in an informative way than just
doing so randomly. This requires knowledge about the specific system that one is trying to
solve. In this case, we were able to use information from many-body perturbation theory
and coupled cluster theory to inform ourselves of a good initial guess.
Finally, Figure 5.10 compares the minimized correlation energy between the two ini-
tialization cases, showing that the MBPT informed (5.96) initial parameterization leads to
more accurate predictions of the ground state correlation energy.
5.3.5 Results
To start, we compare the performances of the classical solution (pCCD) with the
quantum solution (UpCCD via VQE). In Figure 5.11 we see such a comparison for the
101
Figure 5.9: The number of iterations required to minimize the correlation energy (averaged
over 10 trials) are compared for the pairing model with 𝑃 = 4 energy levels and 𝑁 = 2
pairs of particles. itr_rand and itr_ia are the number of iterations required to minimize the
correlation energy for random initial parameters and initial parameters informed by MBPT
(5.96), respectively.
Figure 5.10: VQE calculated ground state correlation energies compared between the case
of random initial parameterization (E_calc_rand) and the case of MBPT informed (5.96)
initial parameterization (E_calc_ia).
pairing model with 𝑃 = 4 energy levels, 𝑁 = 2 pairs, linearly increasing singular particle
energies 𝑑 𝑝 = 𝑝, and constant pairing strength 𝑔 𝑝𝑞 = 𝑔. The VQE results were computed
via a noiseless simulation on a classical computer. We see that for small values of 𝑔 the
102
Figure 5.11: Pairing strength vs ground state correlation energy obtained through exact
diagonalization (blue line), the variational quantum eigensolver (red dots) and pair coupled
cluster doubles theory (green dots) for the pairing model with 𝑃 = 4 energy levels, 𝑁 = 2
pairs, linearly increasing singular particle energies 𝑑 𝑝 = 𝑝, and constant pairing strength
𝑔𝑖𝑎 = 𝑔.
two methods perform relatively similarly. However, as the magnitude of 𝑔 grows, we can
clearly see that the quantum solution gives a better estimate of the ground state correlation
energy than the classical solution.
Additionally, we test our VQE ansatz on a non-constant pairing strength pairing model.
Here, we choose the separable function 𝑔𝑖𝑎 = 𝑔(𝑖𝑎/(max (𝑖) max (𝑎))) which is a more
realistic approximation to real-world nuclei whose nuclear forces are often modeled to be
separable. The function is divided by the maximums of the energy level indices 𝑖 and 𝑗 so
that the values of 𝑔 don’t vary too wildly. In Figure 5.12 we see how pCCD performs well
for small absolute values of 𝑔 but fails for larger values, unlike VQE which gets close to
the true correlation energy values for all values of 𝑔.
103
Figure 5.12: Pairing strength vs ground state correlation energy obtained through exact
diagonalization (blue line), the variational quantum eigensolver (red dots) and pair coupled
cluster doubles (green dots) for the pairing model with 𝑃 = 4 energy levels, 𝑁 = 2
pairs, linearly increasing singular particle energies 𝑑 𝑝 = 𝑝, and constant separable pairing
strength 𝑔𝑖𝑎 = 𝑔(𝑖𝑎/(max(𝑖)(max(𝑎)).
5.3.6 Iterative Quantum Excited States Algorithm
The following is a novel method to compute the eigenstates of a Hamiltonian and their
corresponding energies. Let |𝜓 𝑘 i represent the 𝑘 th eigenstate of the Hamiltonian 𝐻 with
energy 𝐸 𝑛 . That is 𝐻 |𝜓𝑛 i = 𝐸 𝑛 |𝜓𝑛 i. Assume that |𝜓 𝑘 i can be parameterized by the set of
angles 𝜃 𝑘 , that is |𝜓 𝑘 i = |𝜓(𝜃 𝑘 )i. This algorithm is iterative, meaning that to find 𝐸 𝑘 , one
must first use the algorithm to compute 𝐸 0 through 𝐸 𝑘 and 𝜃 0 through 𝜃 𝑘 . Assuming that
one has already done this, recall that any state |𝜓(𝜃)i can be expanded in terms of the 𝑁
eigenstates of 𝐻. That is
Õ𝑁
|𝜓(𝜃)i = h𝜓(𝜃 𝑛 )|𝜓(𝜃)i |𝜓(𝜃 𝑛 )i . (5.100)
𝑛=0
Then, note that the expectation value of 𝐻 of an arbitrary state |𝜓(𝜃)i can be written as
h𝜓(𝜃)|𝐻|𝜓(𝜃)i (5.101)
104
Õ𝑁
= h𝜓(𝜃)|𝜓(𝜃 𝑚 )i h𝜓(𝜃 𝑛 )|𝜓(𝜃)i h𝜓(𝜃 𝑚 )| 𝐻 |𝜓(𝜃 𝑛 )i (5.102)
𝑛,𝑚=0
Õ𝑁
= 𝐸 𝑛 h𝜓(𝜃)|𝜓(𝜃 𝑚 )i h𝜓(𝜃 𝑛 )|𝜓(𝜃)i h𝜓(𝜃 𝑚 )|𝜓(𝜃 𝑛 )i (5.103)
𝑛,𝑚=0
Õ𝑁
= 𝐸 𝑛 h𝜓(𝜃)|𝜓(𝜃 𝑚 )i h𝜓(𝜃 𝑛 )|𝜓(𝜃)i 𝛿𝑛𝑚 (5.104)
𝑛,𝑚=0
Õ𝑁
= 𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃)i| 2 , (5.105)
𝑛=0
which allows
𝑘−1
Õ
h𝜓(𝜃 𝑘 )|𝐻|𝜓(𝜃 𝑘 )i − 𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 (5.106)
𝑛=0
Õ𝑁 𝑘−1
Õ
2
= 𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| − 𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 (5.107)
𝑛=0 𝑛=0
Õ𝑁
= 𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 (5.108)
𝑛=𝑘
Õ 𝑁
≥𝐸 𝑘 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 (5.109)
𝑛=𝑘
" 𝑘−1
#
Õ
=𝐸 𝑘 1 − |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 , (5.110)
𝑛=0
|h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 = 1. This implies that
Í𝑁
since 𝑛=0
𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2
Í 𝑘−1
h𝜓(𝜃 𝑘 )|𝐻|𝜓(𝜃 𝑘 )i − 𝑛=0
𝐸 𝑘 = min . (5.111)
|h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2
Í 𝑘−1
𝜃𝑘 1 − 𝑛=0
The denominator can cause one trouble as small variations near zero would cause wild
jumps in 𝐸 𝑘 . To get around this, we can instead minimize the following function
𝑘−1
Õ
𝐸 𝑘 = min h𝜓(𝜃 𝑘 )|(𝐻 − Δ)|𝜓(𝜃 𝑘 )i − 𝐸 𝑛 |h𝜓(𝜃 𝑛 )|𝜓(𝜃 𝑘 )i| 2 , (5.112)
𝜃𝑘
𝑛=0
105
where here Δ is a large positive number which hopefully shifts the entire energy spectrum
of 𝐻 negative. Note that to apply this algorithm, one must be able to calculate the following
two types of quantities:
• The expectation value of 𝐻 in the 𝑖 th state |𝜓(𝜃 𝑖 )i:
h𝜓(𝜃 𝑖 )|𝐻|𝜓(𝜃 𝑖 )i , (5.113)
• The absolute square of the overlap between two states |𝜓(𝜃 𝑖 )i and |𝜓(𝜃 𝑗 )i:
2
𝜓(𝜃 𝑖 ) 𝜓(𝜃 𝑗 ) . (5.114)
The process to calculate the first quantity, the expectation value of 𝐻, is explained in
section 3.4.2. The technique to efficiently calculate the second quantity, the absolute square
of the overlap between two states, is explained in Appendix E.
As the algorithm is iterative, one starts with the case 𝑘 = 0, in which case the algorithm
reduces to the variational quantum eigensolver for the ground state energy
min h𝜓(𝜃 0 )|𝐻|𝜓(𝜃 0 )i = 𝐸 0 . (5.115)
𝜃0
After obtaining 𝜃 0 and 𝐸 0 through minimization, one uses these quantities to apply the
algorithm for 𝑘 = 1.
min h𝜓(𝜃 1 )|(𝐻 − Δ)|𝜓(𝜃 𝑘 )i − 𝐸 0 |h𝜓(𝜃 0 )|𝜓(𝜃 0 )i| 2 = 𝐸 1 . (5.116)
𝜃1
After minimizing, one obtains 𝜃 1 and 𝐸 1 . One then continues this process for 𝑘 = 2, ..., 𝑁,
ultimately resulting in knowledge of the parameters 𝜃 0 , ..., 𝜃 𝑁 that paramaterize all of the
eigenstates of 𝐻 and all of their corresponding energies 𝐸 0 , ..., 𝐸 𝑁 . Thus the eigenspectrum
106
Figure 5.13: Comparison of correlation energies calculated using the iterative quantum
excited states algorithm and direct diagonalization for the pairing model with 𝑃 = 4 energy
levels, 𝑁 = 2 pairs, linearly increasing singular particle energy 𝑑 𝑝 = 𝑝, and constant
pairing strength 𝑔𝑖𝑎 = 𝑔.
of 𝐻 can be fully calculated. We have tested this algorithm for the first two states of the
pairing model with (𝑃, 𝑁) = (4, 2). In Figure 5.13, one can see that while the algorithm
estimates the ground state quite well, the estimation for the first excited states grows worse
for larger values of 𝑔. We hypothesize that this is due to the accumulation of errors
present in this algorithm (5.112) for large values of 𝑔 as such values make it harder for the
minimization algorithm to succeed. If the algorithm’s estimate for the ground state energy
and/or the ground state are slightly off, this error will become compounded in estimations
for the energies of higher energy levels.
5.3.7 Multi-Configuration Method
We have observed that in the constant 𝑔 case, classical pCCD (and to a lesser but still
significant extent) VQE with the UpCCD ansatz, have a harder and harder time finding
107
the true ground state as 𝑔 grows. This is because the eigenfunction of the ground state
becomes more and more entangled as 𝑔 increases and begins to overtake the strength of the
single-particle energies 𝑑 𝑝 . Such entangled states heuristically take a larger circuit depth
to employ. In the limiting case of 𝑔 → ∞, the one-body term of 𝐻 can be disregarded,
leading to a Hamiltonian that mixes states and yet is not state dependent (𝑔 is constant).
Heuristically, such a Hamiltonian would have, as its ground state, the equal superposition
of all possible states, due to symmetry. Such a state is called a Dicke state. Recall that the
Dicke state [23] has the following recursive form
r r
𝑛 𝑘 𝑛−1 𝑛 − 𝑘 𝑛−1
𝐷𝑘 = 𝐷 𝑘−1 |1i + 𝐷𝑘 |0i . (5.117)
𝑛 𝑛
We will now prove that the Dicke state 𝐷 𝑛𝑘 is indeed an eigenvector of 𝑉 (the limiting
case of 𝐻 where 𝑔 → ∞) where
Õ𝑛
𝑉𝑛 = 𝑔 𝐴†𝑝 𝐴𝑞 . (5.118)
𝑝≠𝑞=1
Specifically
𝑉𝑛 𝐷 𝑛𝑘 = 𝑔𝑘 (𝑛 − 𝑘) 𝐷 𝑛𝑘 . (5.119)
The proof is by induction. First, we prove the base case (𝑛, 𝑘) = (2, 1).
𝑔
𝑉2 𝐷 21 = √ ( 𝐴0† 𝐴1 + 𝐴1† 𝐴0 )(|10i + |01i) = 𝑔 𝐷 21 . (5.120)
2
Next, we assume the induction hypothesis
𝑉𝑛 𝐷 𝑛−1
𝑘 = 𝑔𝑘 (𝑛 − 𝑘 − 1) 𝐷 𝑛−1𝑘 , (5.121)
and use it to prove the statement as follows
𝑛
r r !
Õ 𝑘 𝑛 − 𝑘
𝑉𝑛 𝐷 𝑛𝑘 = 𝑔 𝐴†𝑝 𝐴𝑞 ® 𝐷 𝑛−1
𝑘−1 |1i + 𝐷 𝑛−1 |0i
© ª
𝑘 (5.122)
𝑛 𝑛
« 𝑝≠𝑞=1 ¬
108
𝑛−1 †
Õ 𝑛−1 𝑛−1
†©
Õ ª ©Õ † ª
=𝑔 𝐴 𝑝 𝐴 𝑞 + 𝐴𝑛 𝐴𝑞 ® + 𝐴 𝑝 ® 𝐴𝑛
« 𝑞=1 ¬ « 𝑝=1 ¬#
𝑝≠𝑞=1
"
r r
𝑘 𝑛−1 𝑛 − 𝑘 𝑛−1
× 𝐷 𝑘−1 |1i + 𝐷𝑘 |0i (5.123)
𝑛 𝑛
𝑛−1
r r !
© Õ † ª 𝑘 𝑛−1 𝑛 − 𝑘 𝑛−1
= 𝑔 𝐴 𝑝 𝐴𝑞 ® 𝐷 |1i + 𝐷𝑘 |0i
𝑝≠𝑞=1
𝑛 𝑘−1 𝑛
« ¬
𝑛−1
r 𝑛−1
r
† ©Õ 𝑛 − 𝑘 𝑛−1
Õ
† 𝑘 𝑛−1
+ 𝑔 𝐴𝑛 |0i + 𝐷 𝑘−1 |1i
ª © ª
𝐴𝑞 ® 𝐷𝑘 𝐴 𝑝 ® 𝐴𝑛 (5.124)
𝑛 𝑛
« 𝑞=1 ¬ r « 𝑝=1 ¬
r
𝑘 𝑛−1 𝑛 − 𝑘 𝑛−1
= (𝑘 − 1)(𝑛 − 𝑘) 𝐷 𝑘−1 |1i + 𝑘 (𝑛 − 𝑘 − 1) 𝐷𝑘 |0i
𝑛 𝑛
r r
𝑘 𝑛−1 𝑛 − 𝑘 𝑛−1
+ (𝑛 − 𝑘) 𝐷 𝑘−1 |1i + 𝑘 𝐷𝑘 |0i (5.125)
𝑛 𝑛 !
r r
𝑘 𝑛−1 𝑛 − 𝑘 𝑛−1
= 𝑘 (𝑛 − 𝑘) 𝐷 |1i + 𝐷𝑘 |0i (5.126)
𝑛 𝑘−1 𝑛
= 𝑘 (𝑛 − 𝑘) 𝐷 𝑛𝑘 . (5.127)
In this proof we have used the following lemmas:
Õ 𝑛 p
𝐴†𝑝 𝐷 𝑛𝑘 = (𝑛 − 𝑘)(𝑘 + 1) 𝐷 𝑛𝑘+1 , (5.128)
𝑝=1
Õ 𝑛 p
𝐴 𝐷 𝑛𝑘 = 𝑘 (𝑛 − 𝑘 + 1) 𝐷 𝑛𝑘−1 , (5.129)
𝑝=1
which themselves are proven inductively in Appendix D.
We can see how close the actual ground state of the pairing model (with 𝑑 𝑝 = 𝑝) is
to the Dicke state as 𝑔 grows by plotting their overlap squared. In Figure 5.14) we see
such a plot for various energy levels 𝑝 (with 𝑛 = b 𝑝/2c). The plot shows that by the time
𝑔 reaches about 2.5, the overlap squared of the two states is above 0.9 for all values of 𝑝
109
Figure 5.14: Plot of the overlap squared between the actual ground state and corresponding
Dicke state for pairing models with various values of 𝑝 and 𝑛 = b 𝑝/2c over increasing
values of constant pairing strength.
considered. It is also clear that the ground states for smaller values of 𝑝 become closer to
𝑝
the Dicke state 𝐷 𝑛 at a faster rate than for larger values of 𝑝.
This suggests that, for sufficiently large 𝑔, one may consider the following ansatz for
the pairing model: Use the deterministic [9] or variational (chapter 8) method to initialize
𝑝
one’s quantum circuit to the Dicke state 𝐷 𝑛 for a pairing model with 𝑝 energy levels
and 𝑛 pairs of fermions. Then, slowly increase the number of layers of variational gates
in a "brick wall" fashion (6.1), running VQE on the ansatz each time, until the result no
longer substantially improves. These extra variational gates are there to slightly perturb
the Dicke state to hopefully approximate the true ground state of the pairing model. This
is a so-called "adaptive" ansatz as it can change throughout the process of VQE. We tested
such an ansatz for the pairing model with 𝑃 = 4 energy levels and 𝑁 = 2 pairs of fermions,
110
whose circuit representation is given below
|0i (5.130)
𝐴(𝜃 0 )
|0i
𝑈24 𝐴(𝜃 2 )
|0i
𝐴(𝜃 1 )
|0i
where the gate 𝑈24 prepares the corresponding Dicke state:
𝑈24 |0000i = 𝐷 42 , (5.131)
and 𝐴 is defined the same as the two-qubit operators for the UpCCD ansatz (5.94) except
that the their parameters 𝜃 𝑖 (for 𝑖 = 0, 1, 2) are completely free. That is
𝐴(𝜃) = exp {𝑖𝜃 (𝑋𝑎𝑌𝑖 − 𝑌𝑎 𝑋𝑖 )} , (5.132)
which, as analyzed previously, restricts the minimization algorithm’s search to the correct
Hamming weight subspace. The results of using such an ansatz for VQE can be seen in
Figures 5.15 and 5.16. The first shows a comparison between the VQE estimated and exact
correlation energies while the second shows the relative error between the two from the
first plot. As expected, this multi-configuration ansatz described above works better for
larger absolute values of 𝑔, because we start in a state (the Dicke state) that has a significant
overlap with the ground state. We name the ansatz the multi-configuration ansatz as it is
initialized to the superposition of multiple initial configurations of the fermion pairs (or,
bit-string states).
In future work, it would be of interest to explore if one can get away with preparing a
Dicke like state (equal superposition of bit-string states but with incorrect phases, such as
those prepared in chapter 8) or even partial Dicke states (the equal superposition of some of
111
Figure 5.15: Plot of VQE estimated versus exact correlation energies for the pairing model
with 𝑃 = 4 energy levels and 𝑁 = 2 pairs of fermions using the multi-configuration ansatz.
Figure 5.16: Plot of relative error between VQE estimated and exact correlation energies
for the pairing model with 𝑃 = 4 energy levels and 𝑁 = 2 pairs of fermions using the
multi-configuration ansatz.
112
the bit-string states that make up an entire Dicke states). One could also explore this ansatz
for a larger number of energy levels 𝑃 and different formulas of the single particle energies
𝑑 𝑝 . Finally, on might consider using the first few layers of the UpCCD ansatz to try to
minimize the pairing model with 𝑑 𝑝 = 0 which should get one close to a Dicke state. Then
add in the rest of the layers and minimize to the pairing model that one desires, initializing
the initial layers to the parameters that the minimization algorithm found approximated the
Dicke state.
5.4 Conclusion
In this section, we introduced the pairing model, a toy model for many-body nuclear
physics upon which many new techniques are tested. First, we explored ways to solve the
problem on a classical computer, including through the use of many-body perturbation
theory and coupled cluster theory. These two methods served to help us in the next section
by informing a good set of initial guesses for our variational circuit. In the aforementioned
section, we walked through how to map the pairing model Hamiltonian and an extension
of the unitary coupled cluster ansatz from pair fermionic operators to spin operators via an
extension to the Jordan-Wigner transformation. We then presented our results of applying
VQE to the pairing model and bench-marked it against classical coupled cluster theory. We
went on to extend VQE in two novel ways: first, we introduced the novel iterative quantum
excited states algorithm to search for the energy levels of excited states of the pairing
model. We then introduced the novel multi-configurational ansatz for pairing models with
large constant pairing strengths 𝑔. In the future, one might like to compare our excited
states algorithms to similar algorithms and also test various relaxed forms of the multi-
configurational ansatz that involve the initialization of Dicke-like states. Finally, one may
113
wish to apply the VQE solution to the pairing problem to approximate an actual nucleus
that has strong pairing interactions such as ones with doubly-magic shells with valence
electrons of a single type. Here, we have given a springboard off of which future research
may be accomplished as quantum computing tackles ever more complex many-body nuclear
physics systems.
114
CHAPTER 6
COLLECTIVE NEUTRINO OSCILLATIONS
6.1 Introduction
The next system we will consider is that of collective neutrino oscillations, which
has Hamiltonian that is mathematically similar to that of the previous pairing models
considered. Instead of using quantum algorithms to determine the energy spectrum of
the Hamiltonian, as done in the two previous cases, we will instead use them to simulate
the time-evolution of the system and measure its entanglement properties. Neutrinos are
nuclear particles in the sense that they interact via the weak nuclear force (and gravity). The
motivation for studying this system comes from the flavor evolution of neutrinos in dense
astrophysical environments. It has been pointed out by Pantelone, Raffelt, and Sigl [61,
76] and others that neutrinos can exchange their flavors through forward scattering. If one
starts with an anisotropic initial distribution of energy and/or angle (as found in supernovae,
neutron star mergers, or the early universe), then the neutrino energy flux versus energy and
flavor may be impacted by this non-trivial quantum many-body evolution. This can in turn
affect the dynamics of these environments and other observables, including nucleosynthesis
in the ejected material [25, 18].
Most often, these quantum equations have been treated on the mean-field level by
replacing one of the spin operators in equation (6.1) by its expectation value, yielding a
set of non-linear coupled differential equations. This makes the calculations tractable for
several hundred energies and angles on modern computers [26]. More recently, studies
of neutrino propagation as a quantum many-body problem have appeared, including for
example [10, 35, 72, 62, 71, 16, 68, 67]. These works highlight the importance of
115
understanding the role of quantum correlations, such as entanglement, in order to quantify
beyond mean-field effects in out-of-equilibrium neutrino simulations. A direct solution of
the Schrödinger equation in equation (2.13), for a system of 𝑁 configurations in energy
and angle, incurs a computational cost that is exponential in 𝑁. This has limited early
explorations of the problem to systems with 𝑁 = O (10) neutrinos. An alternative to
reach larger system sizes, explored recently [68, 67], employs a matrix product state
representation for |Φ(𝑡)i which allows one to track the exact time evolution in situations
where entanglement never grows too much. For conditions leading to strong entanglement
instead, simulations on digital/analog quantum computers have the potential to tackle the
full neutrino dynamics while still enjoying a polynomial computational cost in system size
𝑁 [54].
In this section, we explore the time-dependent many-body evolution and entanglement
of neutrinos on a current-generation digital quantum computer. In section 6.2 we introduce
in more detail the SU(2) spin model used to describe collective neutrino oscillations and an
implementation of the time evolution operator appearing in equation (2.13) suitable for an
array of qubits with linear connectivity. We present the results obtained for a small system
with 𝑁 = 4 neutrino amplitudes in section 6.4 and provide a summary and conclusions in
section 6.8.
6.2 Hamiltonian
The Hamiltonian for neutrino flavor evolution in a dense neutrino environment includes
three terms:
1. 𝐻𝑣 | the vacuum mixing that has been determined from solar and accelerator neutrino
experiments [38]
116
2. 𝐻𝑠 | the forward scattering in matter leading to the well known MSW effect [87, 57]
3. 𝐻𝑛 | neutrino-neutrino forward scattering
The first simplification we make is to truncate the neutrino flavors involved from three
to two. That is, we only consider the oscillation of two flavors which, without loss of
generality, we choose to be 𝜈𝑒 and 𝜈𝑥 . Here 𝜈𝑥 describes either 𝜈 𝜇 or 𝜈𝜏 which we assume
evolve similarly. This simplification allows for the neutrino-neutrino interaction 𝐻𝑛 to be
proportional to the dot product 𝜎𝑖 · 𝜎 𝑗 of the SU(2) Pauli matrices describing the different
flavor amplitudes of the two neutrinos
𝑞𝑖 · 𝑞 𝑗
(𝐻𝑛 )𝑖 𝑗 ∝ 1 − 𝜎𝑖 · 𝜎 𝑗 . (6.1)
k𝑞𝑖 kk𝑞 𝑗 k
where 𝑞 𝑘 is the momentum of the 𝑘-th neutrino and 𝜎𝑘 = (𝑋 𝑘 , 𝑌𝑘 , 𝑍 𝑘 ) is the vector of
Pauli operators acting on its amplitude. The neutrino-neutrino interaction can exchange
flavors of two neutrinos and, as can be seen above, has a forward scattering amplitude that
depends on the angle between their momenta. Generalization to the three-flavor case is
straightforward in principle.
We are working in the neutrino flavor basis, whose fermionic operators are related to
those of the mass basis via the rotation
©𝑎 𝑒 ( 𝑝) ª © cos 𝜃 sin 𝜃 ª ©𝑎 1 ( 𝑝) ª
®= ® ®. (6.2)
® ® ®
𝑎 𝑥 ( 𝑝) − sin 𝜃 cos 𝜃 𝑎 2 ( 𝑝)
« ¬ « ¬« ¬
In this basis the vacuum term 𝐻𝑣 includes diagonal contributions describing the mass
differences between different neutrino flavors and an off-diagonal term characterized by a
mixing angle 𝜃 𝑣 , while forward scattering term 𝐻𝑠 is diagonal in the flavor basis.
117
For the simplified two-flavor case studied here, the state of the system can be described
as an amplitude for a neutrino of each energy 𝐸𝑖 (equal to the magnitude of momentum
k𝑞𝑖 k) and direction of momentum (denoted by 𝑞𝑖 ), with 𝛼↑ and 𝛼↓ describing the amplitude
of being in the electron flavor or in the heavy 𝑥 (𝜇 or 𝜏) flavor respectively. These two
amplitudes can be encoded in an SU(2) spinor basis. In this basis, the Hamiltonian can be
written in terms of Pauli operators as the sum of a one-body term, describing both vacuum
oscillations and forward scattering in matter,
1Õ
𝐻1 = (−Δ𝑖 cos 2𝜃 𝑣 + 𝐴) 𝜎𝑖𝑧 + Δ𝑖 sin 2𝜃 𝑣 𝜎𝑖𝑥 , (6.3)
2 𝑖
and a two-body term, coming from the neutrino-neutrino forward-scattering potential 𝑉𝑖 𝑗
from equation (6.1), which takes the following form [62]
Õ
𝐻2 = 𝜂[1 − 𝑞ˆ𝑖 · 𝑞ˆ 𝑗 ]𝜎𝑖 · 𝜎 𝑗 . (6.4)
𝑖< 𝑗
In the one-body term, 𝜃 𝑣 represents the vacuum mixing angle, while the strength is given
by Δ𝑖 = 𝛿𝑚 2 /(2𝐸𝑖 ) with 𝛿𝑚 2 the mass squared difference for neutrinos of different flavor.
The matter potential enters as the diagonal contribution in the one-body term through the
√
constant 𝐴 = 2𝐺 𝐹 𝑛𝑒 , with 𝐺 𝐹 the Fermi coupling constant and 𝑛𝑒 the electron density.
As described in the introduction, the two-body term is a sum over spin-spin interactions
with a coupling depending upon the relative angle between them. The overall strength
depends on the neutrino density as
𝐺𝐹 𝐺 𝐹 𝑛𝜈
𝜂=√ = √ , (6.5)
2𝑉 2𝑁
with 𝑁 the number of neutrino momenta considered, given by the neutrino density 𝑛𝜈 times
the quantization volume 𝑉. We can transform from fermionic flavor operators to flavor
118
isospin operators via the Jordan-Schwinger mapping:
𝐽 𝑝+ = 𝑎 †𝑒 ( 𝑝)𝑎 𝑥 ( 𝑝), (6.6)
𝐽 𝑝− = 𝑎 †𝑥 ( 𝑝)𝑎 𝑒 ( 𝑝), (6.7)
1 †
𝐽 𝑝𝑧 = 𝑎 𝑒 ( 𝑝)𝑎 𝑒 ( 𝑝) − 𝑎 †𝑥 ( 𝑝)𝑎 𝑥 ( 𝑝) , (6.8)
2
which obey the SU(2) commutation relations
[𝐽 𝑝+ , 𝐽𝑞− ] = 2𝛿 𝑝𝑞 𝐽 𝑝𝑧 (6.9)
[𝐽 𝑝𝑧 , 𝐽𝑞± ] = ±𝛿 𝑝𝑞 𝐽 𝑝 ,
𝑝𝑚
(6.10)
and are thus isomorphic to the Pauli-spin matrices. Note that we can transform to Cartesian
coordinates through the definition
𝐽 𝑝± = 𝐽 𝑝𝑥 ± 𝑖𝐽 𝑝
𝑦
(6.11)
The term of the Hamiltonian that describes vacuum oscillations 𝐻𝜈 is given by
!
Õ 𝑚2 𝑚 2
1 †
𝐻𝜈 = 𝑎 1 ( 𝑝)𝑎 1 ( 𝑝) + 2 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝) , (6.12)
𝑝
2𝑝 2𝑝
in the mass basis. To transform it to the Pauli basis, we first subtract it by the following
term
Õ 𝑚2 + 𝑚2
1 2
𝑎 †1 ( 𝑝)𝑎 1 ( 𝑝) + 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝) , (6.13)
𝑝
4𝑝
which can be done without consequence as it is proportional to the identity (for a given
number of particles) since the total number of neutrinos in each momentum mode is
constant. This follows from the fact that forward scattering only exchanges the neutrino’s
momenta. Subtracting said term transforms the vacuum oscillation term (6.12) into
Õ Δ𝑚 2
† †
𝐻𝜈 = 𝑎 2 ( 𝑝)𝑎 2 ( 𝑝) − 𝑎 1 ( 𝑝)𝑎 1 ( 𝑝) , (6.14)
𝑝
4𝑝
119
where Δ2𝑚 = 𝑚 22 − 𝑚 12 . Applying the inverse of the mapping between the flavor and mass
bases (6.2) yields
Õ Δ𝑚 2 h
𝐻𝜈 = (sin 𝜃𝑎 †𝑒 ( 𝑝) + cos 𝜃𝑎 †𝑥 ( 𝑝)) (sin 𝜃𝑎 𝑒 ( 𝑝) + cos 𝜃𝑎 𝑥 ( 𝑝))
𝑝
4𝑝
i
− (cos 𝜃𝑎 †𝑒 ( 𝑝) − sin 𝜃𝑎 †𝑥 ( 𝑝)) (cos 𝜃𝑎 𝑒 ( 𝑝) − sin 𝜃𝑎 𝑥 ( 𝑝)) (6.15)
Õ Δ𝑚 2 h
† †
= sin 2𝜃 𝑎 𝑒 ( 𝑝)𝑎 𝑥 ( 𝑝) + 𝑎 𝑥 ( 𝑝)𝑎 𝑒 ( 𝑝)
𝑝
4𝑝
i
− cos 2𝜃 𝑎 †𝑒 ( 𝑝)𝑎 𝑒 ( 𝑝) − 𝑎 †𝑥 ( 𝑝)𝑎 𝑥 ( 𝑝) , (6.16)
which can be mapped to the Pauli basis via (6.6)-(6.8), resulting in
Õ
𝐻𝜈 = 𝐵 · 𝜎𝑝 , (6.17)
𝑝
where
Δ𝑚 2
𝐵= (sin 2𝜃, 0, − cos 2𝜃). (6.18)
2𝑝
We’ve written the Hamiltonian in terms of Paul operators instead of the SU(2) operators
𝐽 ± and 𝐽 𝑧 as the two are isomorphic. The neutrino-neutrino scattering term is given by
𝐺𝐹 Õ
(1 − cos 𝜙 𝑝𝑞 ) 𝑎 †𝑒 ( 𝑝)𝑎 𝑥 ( 𝑝)𝑎 †𝑥 (𝑞)𝑎 𝑒 (𝑞) + 𝑎 †𝑥 ( 𝑝)𝑎 𝑒 ( 𝑝)𝑎 †𝑒 (𝑞)𝑎 𝑥 (𝑞)
𝐻𝜈𝜈 = √
2𝑉 𝑝𝑞
+ 𝑎 †𝑒 ( 𝑝)𝑎 𝑒 ( 𝑝)𝑎 †𝑒 (𝑞)𝑎 𝑒 (𝑞) + 𝑎 †𝑥 ( 𝑝)𝑎 𝑥 ( 𝑝)𝑎 †𝑥 (𝑞)𝑎 𝑥 (𝑞) ,
(6.19)
which can also be mapped to the Pauli basis via (6.6)-(6.8), resulting in
𝐺𝐹 Õ
𝐻𝜈𝜈 = √ (1 − cos 𝜙 𝑝𝑞 ) 𝜎𝑝+ 𝜎𝑞− + 𝜎𝑞− 𝜎𝑝+ + 𝜎𝑝𝑧 𝜎𝑞𝑧
2𝑉 𝑝𝑞
120
Õ
= 𝐽 𝑝𝑞 𝜎𝑝 · 𝜎𝑞 , (6.20)
𝑝𝑞
with
√
2𝐺 𝐹
𝐽 𝑝𝑞 = (1 − cos 𝜙 𝑝𝑞 ), (6.21)
𝑉
where we’ve disregarded the term 𝑎 †𝑒 ( 𝑝)𝑎 𝑒 ( 𝑝)𝑎 †𝑥 (𝑞)𝑎 𝑥 (𝑞) + 𝑎 †𝑥 ( 𝑝)𝑎 𝑥 ( 𝑝)𝑎 †𝑒 (𝑞)𝑎 𝑒 (𝑞) as it
is proportional to the identity. Note that, again, this is given in terms of Pauli spin matrices
as opposed to angular momentum operators as the two are isomorphic as they obey the
same SU(2) commutation relations. Putting the two terms together, the Hamiltonian for
collective neutrino oscillations becomes
Õ Õ
𝐻= 𝐵 · 𝜎𝑝 + 𝐽 𝑝𝑞 𝜎𝑝 · 𝜎𝑞 , (6.22)
𝑝 𝑝<𝑞
where we’re able to restricted to sum to 𝑝 < 𝑞 as the 𝑝 = 𝑞 term is proportional to the
identity and restricting 𝑝 ≠ 𝑞 to 𝑝 < 𝑞 only picks up a factor of two which can be absorbed
into the constant 𝐽 𝑝𝑞 . We note that the Hamiltonian is similar to the Heisenberg model
except that the two-body term is all-to-all rather than nearest neighbor; that is, it sums over
both 𝑝 and 𝑞 rather than summing over 𝑝 and 𝑞 = 𝑝 + 1. Its coupling strength 𝜂 ∝ 1/𝑁
assures that the energy of the system is extensive. This allows us to obtain a well-defined
many-body solution, in the limit of large numbers of neutrino momenta by extrapolating
in system size 𝑁.
Because NISQ era quantum computers must have a relatively limited circuit depth in
order to not be too noisy, the maximum time to which we can simulate time-evolution
is also limited. Therefore, we consider a test case where the one-body and two-body
interaction terms are set to be similar in magnitude, allowing flavor oscillations to occur
121
rapidly. An example of this case is the environment of order 100km from the surface of a
proto-neutron star in a core collapse supernova. Here, the background matter density has
decreased to the point where its contribution to the Hamiltonian is similar in magnitude to
the neutrino-neutrino forward scattering. The relative angles of neutrino propagation are
fairly small as neutrinos are emitted from a typical proto-neutron star radius of order 10km.
In the neutrino bulb model [26] one further assumes that the evolution in a supernova
depends only on the energy and the angle from the normal. Averaging over the azimuthal
angles results in an average coupling h1 − 𝑞ˆ𝑖 · 𝑞ˆ 𝑗 i = 1 − cos(𝜃 𝑖 ) cos 𝜃 𝑗 .
For our test case, we take a monochromatic neutrino beam with energy 𝐸 𝜈 = 𝛿𝑚 2 /(4𝜂)
and measure energies in units of the two-body coupling 𝜂. In order to avoid the symmetries
introduced by the single angle approximation, we employ an anisotropic distribution of
momentum directions using a simple grid of angles with
| 𝑝 − 𝑞|
𝜙 𝑝𝑞 = arccos(0.9) . (6.23)
𝑁 −1
This is similar to the standard bulb model as the relative couplings 1 − cos 𝜙 𝑝𝑞 are small.
Additionally, we choose 𝜃 = 0.195 so that the one-body and two-body terms are of relative
strength. This leads the parameters to have the following numerical values:
p
𝐵= 1− 0.9252 , 0, −0.925 (6.24)
| 𝑝 − 𝑞|
𝐽 𝑝𝑞 = 1 − cos arccos 0.9 . (6.25)
𝑁 −1
6.3 Connection to the Pairing Model
Before we go any further, we’d like to show here how collective neutrino oscillations can
be viewed as a pairing model, thus justifying its use as an application of the pairing model.
122
This connection can be seen by writing the collective neutrino oscillation Hamiltonian in
the mass basis. To do so, we introduce the mass isospin operators
𝐾 𝑝+ = 𝑎 †1 ( 𝑝)𝑎 2 ( 𝑝) (6.26)
𝐾 𝑝− = 𝑎 †2 ( 𝑝)𝑎 1 ( 𝑝) (6.27)
1 †
𝐾 𝑝𝑧 = 𝑎 1 ( 𝑝)𝑎 1 ( 𝑝) − 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝) , (6.28)
2
which are analogous to the flavor isospin operators (6.6-6.8). This allows the vacuum
oscillation term (6.14) to be readily identified as
Õ Δ𝑚 2
𝐻𝜈 = 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝) − 𝑎 †1 ( 𝑝)𝑎 1 ( 𝑝)
𝑝
4𝑝
Õ
= 𝜔 𝑝 𝐾 𝑝𝑧 , (6.29)
𝑝
where
Δ𝑚 2
𝜔=− . (6.30)
2𝑝
In order to deal with the neutrino neutrino interaction term (6.17), we must find the mapping
between flavor isospin operators and mass isospin operators which follow from the mapping
between flavor fermionic operators and mass fermionic operators (6.2):
𝐽 𝑝+ = 𝑎 †𝑒 ( 𝑝)𝑎 𝑥 ( 𝑝)
= [cos 𝜃𝑎 †1 ( 𝑝) + sin 𝜃𝑎 †2 ( 𝑝)] [cos 𝜃𝑎 2 ( 𝑝) − sin 𝜃𝑎 1 ( 𝑝)]
1
= cos2 𝜃𝑎 †1 ( 𝑝)𝑎 2 ( 𝑝) − sin2 𝜃𝑎 †2 ( 𝑝)𝑎 1 ( 𝑝) − sin 2𝜃 [𝑎 †1 ( 𝑝)𝑎 1 ( 𝑝) − 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝)]
2
= cos2 𝜃𝐾 𝑝+ − sin2 𝜃𝐾 𝑝− − sin 2𝜃𝐾 𝑧 , (6.31)
𝐽 𝑝− = 𝑎 †𝑒 ( 𝑝)𝑎 𝑥 ( 𝑝)
123
= [cos 𝜃𝑎 †2 ( 𝑝) − sin 𝜃𝑎 †1 ( 𝑝)] [cos 𝜃𝑎 1 ( 𝑝) + sin 𝜃𝑎 2 ( 𝑝)]
1
= cos2 𝜃𝑎 †2 ( 𝑝)𝑎 1 ( 𝑝) − sin2 𝜃𝑎 †1 ( 𝑝)𝑎 2 ( 𝑝) − sin 2𝜃 [𝑎 †1 ( 𝑝)𝑎 1 ( 𝑝) − 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝)]
2
= cos2 𝜃𝐾 𝑝− − sin2 𝜃𝐾 𝑝+ − sin 2𝜃𝐾 𝑧 , (6.32)
1 †
𝐽 𝑝𝑧 = 𝑎 𝑒 ( 𝑝)𝑎 𝑒 ( 𝑝) − 𝑎 †𝑥 ( 𝑝)𝑎 𝑥 ( 𝑝)
2
1
= {[cos 𝜃𝑎 †1 ( 𝑝) + sin 𝜃𝑎 †2 ( 𝑝)] [cos 𝜃𝑎 1 ( 𝑝) + sin 𝜃𝑎 2 ( 𝑝)]
2
− [cos 𝜃𝑎 †2 ( 𝑝) − sin 𝜃𝑎 †1 ( 𝑝)] [cos 𝜃𝑎 2 ( 𝑝) − sin 𝜃𝑎 1 ( 𝑝)]}
1
= {cos 2𝜃 [𝑎 †1 ( 𝑝)𝑎 1 ( 𝑝) − 𝑎 †2 ( 𝑝)𝑎 2 ( 𝑝)] + sin 2𝜃 [𝑎 †1 ( 𝑝)𝑎 2 ( 𝑝) − 𝑎 †2 ( 𝑝)𝑎 1 ( 𝑝)]}
2
1
= cos 2𝜃𝐾 𝑝𝑧 + sin 2𝜃 (𝐾 𝑝+ + 𝐾 𝑝− ), (6.33)
2
which can be expressed succinctly in matrix form, for reference, as
© 𝐽 𝑝+ ª © cos2 𝜃 − sin2 𝜃 − sin 2𝜃 ª © 𝐾 𝑝+ ª
® ® ®
−®
𝐽 𝑝 ® = − sin 𝜃 cos 𝜃 − sin 2𝜃 ® 𝐾 𝑝− ® .
2 2 ® ®
(6.34)
® ® ®
® ® ®
𝑧 1 1
𝐽𝑝 sin 2𝜃 sin 2𝜃 cos 2𝜃 𝐾𝑧
« ¬ «2 2 ¬ « 𝑝¬
This implies that the dot product of the total flavor isospin operators is equal to the dot
product of the total mass isospin operators:
1 + −
𝐽 𝑝 · 𝐽𝑞 = (𝐽 𝐽 + 𝐽 𝑝− 𝐽𝑞+ ) + 𝐽 𝑝𝑧 𝐽𝑞𝑧
2 𝑝 𝑞
© 𝐽 𝑝− ª
1 + − 𝑧 + ®®
= 𝐽 𝐽 𝐽 𝑝 𝐽 𝑝 ®®
2 𝑝 𝑝
®
2𝐽 𝑧
« 𝑝¬
© cos2 𝜃 − sin2 𝜃 1
2 sin 2𝜃 ª © cos2 𝜃 − sin2 𝜃 − sin 2𝜃 ª ©𝐾 𝑝− ª
1 + ® ® ®
= −
𝐾 𝑝 𝐾 𝑝 𝐾 𝑝 − sin2 𝜃 cos2 𝜃 1 ® 2 cos 𝜃 − sin 2𝜃 ®® 𝐾 𝑝+ ®®
2 ® ®
2 sin 2𝜃 ® − sin 𝜃
𝑧 ®
2
® ® ®
− sin 2𝜃 − sin 2𝜃 cos 2𝜃 sin 2𝜃 sin 2𝜃 2 cos 2𝜃 𝐾 𝑝𝑧
« ¬« ¬« ¬
124
©1 0 0ª ©𝐾 𝑝− ª
1 + ® ®
= 𝐾 𝑝 𝐾 𝑝 𝐾 𝑝 0 1 0®® 𝐾 𝑝+ ®®
− 𝑧 ® ®
2
® ®
0 0 2 𝐾 𝑝𝑧
« ¬« ¬
1 + −
= (𝐾 𝑝 𝐾𝑞 + 𝐾 𝑝− 𝐾𝑞+ ) + 𝐾 𝑝𝑧 𝐾𝑞𝑧 (6.35)
2
= 𝐾 𝑝 · 𝐾𝑞 . (6.36)
Thus, the neutrino-neutrino interaction term (6.20) can be written in terms of total mass
isospin operators as
Õ
𝐻𝜈𝜈 = 𝐽 𝑝𝑞 𝐾 𝑝 · 𝐾𝑞 . (6.37)
𝑝𝑞
Putting the terms (6.29 - 6.37) together gives us the Hamiltonian in terms of mass isospin
operators
Õ Õ
𝐻= 𝜔 𝑝 𝐾 𝑝𝑧 + 𝐽 𝑝𝑞 𝐾 𝑝 · 𝐾𝑞 . (6.38)
𝑝 𝑝𝑞
Consider now the single-angle approximation, where we assume that neutrinos travel-
ing in different directions undergo the same flavor evolution, implying that the term
Í Í
𝑝𝑞 cos 𝜙 𝑝𝑞 𝐽 𝑝 · 𝐽𝑞 (and therefore also 𝑝𝑞 cos 𝜙 𝑝𝑞 𝐾 𝑝 · 𝐾 𝑞 ) averages to zero, which (after
recalling 𝐽 𝑝𝑞 = 𝜇(1 − cos 𝜙 𝑝𝑞 )) leads to the Hamiltonian
Õ Õ
𝐻= 𝜔 𝑝 𝐾 𝑝𝑧 + 𝜇 𝐾 𝑝 · 𝐾𝑞 , (6.39)
𝑝 𝑝𝑞
which can be written as
Õ Õ
𝑦 𝑦
𝐻= 𝜔 𝑝 𝐾 𝑝𝑧 + 2𝜇 (𝐾 𝑝𝑥 · 𝐾𝑞𝑥 + 𝐾 𝑝 · 𝐾𝑞 ), (6.40)
𝑝 𝑝<𝑞
125
Í Í 𝑦 𝑦 Í
𝐾 𝑝𝑧 · 𝐾𝑞𝑧 and 𝜇 𝑝=𝑞 (𝐾 𝑝 · 𝐾𝑞𝑥 + 𝐾 𝑝 · 𝐾𝑞 ) = 2𝜇
where we’ve dropped the terms 𝜇 𝑥
𝑝𝑞 𝑝 𝐼𝑝
as they are both proportional to the identity (for a fixed number of neutrinos) and therefore
have no effect on the time-evolution of the system. Recall now the pairing Hamiltonian in
𝑝
terms of Pauli spin operators (5.86) which in the case of constant pairing strength (𝑔𝑞 = 𝑔)
we can write as
𝑃
1Õ 1 Õ
𝐻=− (2𝑑 𝑝 + 𝑔)𝑍 𝑝 + 𝑔 (𝑋 𝑝 𝑋𝑞 + 𝑌𝑝𝑌𝑞 ), (6.41)
2 𝑝=1 2 𝑝,𝑞=1
𝑝<𝑞
1 Í𝑃
by dropping the term 2 𝑝=1 (2𝑑 𝑝 + 𝑔)𝐼 𝑝 as it is proportional to the identity. With this,
we can see that, since the mass isospin operators and Pauli-spin operators are isomorphic
(share the same commutation relations), the collective neutrino oscillation Hamiltonian in
the single-angle approximation (6.40) is equivalent (up to terms proportional to the identity)
to the pairing Hamiltonian with constant pairing strength (6.41) if we set 𝑑 𝑝 = 𝜔 𝑝 − 𝑔/2
and 𝑔 = 4𝜇. This implies that the two systems evolve identically in time.
6.4 Time Evolution
One of the challenges in implementing the time evolution of the collective neutrino
oscillation Hamiltonian (6.22) on a quantum computer is to find an accurate approximation
to the time evolution operator (2.14)
𝑈 (𝑡) = exp{−𝑖𝐻𝑡}, (6.42)
that can be decomposed efficiently into quantum gates [54]. In this work, we accomplish
this by using a first-order Trotter-Suzuki decomposition [78] of the time evolution operator.
This decomposition requires partitioning the Hamiltonian. A naive partitioning would be
to simply keep the Hamiltonian (6.22) written as is which would lead to the following
126
approximation for the time-evolution operator
Ö 𝑁 Ö 𝑁
𝑈1 (𝑡) = 𝑒 −𝑖𝑡𝐵·𝜎𝑝 𝑒 −𝑖𝑡𝐽 𝑝𝑞 𝜎𝑝 ·𝜎𝑞 . (6.43)
𝑝=1 𝑝<𝑞=1
Instead, we choose the pair propagation partition, in which we rewrite the Hamiltonian
(6.22) as a strictly two-body term
Õ𝑁
𝐻= ℎ 𝑝𝑞 , (6.44)
𝑝<𝑞
where
1
ℎ 𝑝𝑞 = 𝐵 · (𝜎𝑝 + 𝜎𝑞 ) + 𝐽 𝑝𝑞 𝜎𝑝 · 𝜎𝑞 , (6.45)
𝑁 −1
which is permitted, as
Õ𝑛 Õ 𝑛 Õ 𝑛
(𝜎𝑝 + 𝜎𝑞 ) = (𝜎𝑝 + 𝜎𝑞 )
𝑝<𝑞 𝑝=1 𝑞=𝑝+1
Õ 𝑛 Õ 𝑛 Õ 𝑛 Õ 𝑞−1
= 𝜎𝑝 + 𝜎𝑞
𝑝=1 𝑞=𝑝+1 𝑞=2 𝑝=1
Õ 𝑛 Õ 𝑛
= (𝑛 − 𝑝)𝜎𝑝 + (𝑞 − 1)𝜎𝑞
𝑝=1 𝑞=1
Õ 𝑛
= [(𝑛 − 𝑝) + ( 𝑝 − 1)] 𝜎𝑝
𝑝=1
Õ 𝑛
= (𝑛 − 1) 𝜎𝑝 . (6.46)
𝑝=1
This leads to the pair-propagation time-evolution operator approximation 𝑈2 , defined as
Ö𝑁
𝑈2 (𝑡) = 𝑢 𝑝𝑞 (6.47)
𝑝<𝑞
where
𝑢 𝑝𝑞 = 𝑒 −𝑖𝑡ℎ 𝑝𝑞 , (6.48)
127
which is correct up to additive error 𝜖 = O (𝑡 2 ). This partitioning is motivated by past
experience with the Euclidean version of this evolution operator in quantum Monte Carlo
which suggests that this partitioning yields a better approximation to the time-evolution
operator 𝑈 (𝑡) (see eg. [15, 14]). The main reason we choose this partitioning, however, is
that it has better error scaling than the original partitioning 𝑈1 , as detailed below. While
asymptotic scaling of the approximation error 𝜖 is quadratic in the time-step 𝑡 for both
approximations [78], the pair approximation is expected to perform better in practice for
cases where an accurate description of pair evolution is important due, for instance, to
strong cancellations between the one-body and two-body contributions in the Hamiltonian.
In the neutrino case, these situations can occur with appropriate initial conditions so that,
for typical states in the evolution, we have for most pairs that
h𝐾 𝑝𝑞 i + h𝑉𝑝𝑞 i h𝐾 𝑝𝑞 i + h𝑉𝑝𝑞 i (6.49)
where we have used the short-hand
1
h𝐾 𝑝𝑞 i = 𝐵 · h𝜎𝑝 + 𝜎𝑞 i (6.50)
𝑁 −1
h𝑉𝑝𝑞 i = 𝐽 𝑝𝑞 h𝜎𝑝 · 𝜎𝑞 i. (6.51)
Since the difference between the two approximations is not expected to hold for a generic
initial state, standard error measures like the matrix norm of the difference with the exact
propagator
exp (−𝑖𝑡𝐻) − 𝑈1/2 (𝑡) (6.52)
are not expected to capture the effect. This is, in fact, found in practice for our system. In
panel (a) of Figure 6.1 we show the estimate from equation (6.52) for the 𝑁 = 4 neutrino
model considered in this work. This error estimate indicates that the 𝑈1 approximation has
128
a smaller maximum error than 𝑈2 up to long times. We can look at a more direct measure
of accuracy for our specific setup by considering instead the state fidelity
2
𝑓 (𝑡) = hΨ1,2 (𝑡)|Ψ(𝑡)i (6.53)
between the exact state |Ψ(𝑡)i at time 𝑡 and one of its approximations Ψ1,2 (𝑡) , defined as
Ψ1,2 (𝑡) = 𝑈1,2 |Ψ(0)i . (6.54)
We show 𝑓 (𝑡) for both approximations in panel (b) of Figure 6.1. The result here suggest
that instead the pair approximation produces a state with a higher fidelity with the true state
than the simple linear propagator 𝑈1 , especially at relatively long time-steps 𝑡 ∈ [4, 8].
2 1
Error in matrix norm Inversion probability
(a) (c) Neutrino 1 0.8
1.5
0.6
1
U exact
U1 - approx 0.4
0.5 U2 - approx
0.2
1 1
Inversion probability
0.8 Neutrino 2 0.8
State fidelity
0.6 0.6
0.4 0.4
0.2 0.2
(b) (d)
0 0
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12
−1 −1
Time t [η ] Time t [η ]
Figure 6.1: Panel (a) shows the error in matrix 2-norm equation (6.52) of the two approx-
imations 𝑈1 and 𝑈2 described in the text. Panel (b) shows the state fidelity and the right
panels show results for the inversion probability 𝑃inv (𝑡). Panel (c) is for neutrino one while
panel (d) is for neutrino 2.
Finally, since we are mostly interested in flavor observables diagonal in the computa-
tional basis, we also show a direct comparison of the inversion probability for two out of
129
the 𝑁 = 4 neutrinos using both approximations and the exact propagator (panels (c) and
(d)). The details of how we compute the inversion probability are forthcoming but these
selected results were included here first to show more clearly that the pair approximation
allows us to correctly describe the evolution of flavor for substantially longer times than the
𝑈1 approximation. The results reported here do depend on the specific choice of ordering
of qubits in the time evolution layers shown in Figure 6.2. In both the present analysis
and the simulation results we used the best ordering which we empirically found to be
(1, 3, 2, 4) as one would’ve expected based on the initial state and the criterion equation
(6.49) above. A more rigorous discussion of the relative accuracy between the canonical
first order and the pair approximation, together with the effect of ordering choices, could
be explored in future work.
Because our Hamiltonian (6.44) sums over all 𝑝 < 𝑞, a naive implementation would
require either a device with all-to-all qubit-connectivity (like a trapped ion systems [58])
or an extensive use of the SWAP gate (3.37). Recall that the SWAP operation exchanges
the states of the two qubits upon which it acts. It can therefore be used to bring the states
of any pair of qubits next to one another in the physical qubit space, allowing one to apply
two-qubit interactions between them on a device with linear qubit-connectivity. This would
naively require a sequence of order 𝑁 SWAP gates. However, we will show that, since we
need to apply all possible pair interactions, it is actually possible to carry out a complete
Trotter step of the time-evolution operator (6.47) without incurring any overhead due to the
application of the SWAP operations. The scheme is inspired by the more general fermionic
swap network construction presented in [51].
We illustrate this idea using the diagram shown in Figure 6.2 for a simple case with
130
𝑁 = 4 neutrinos. Starting from the initial state on the left, we first apply the unitaries 𝑢 𝑝𝑞
from equation (6.47) to the odd bonds: for the 𝑁 = 4 case, these are the bonds between the
(1, 2) and (3, 4) pairs of qubits. Before moving to the next pairs, we also apply a SWAP
operation to the same pairs upon which we just acted. The resulting unitary operation is
denoted as a double line joining circles (qubits) in Figure 6.2 and the net effect is that at the
next step the qubits that have interacted get interchanged. As any two-qubit unitary can be
efficiently decomposed into a sequence of gates involving three CNOTs and 15 single qubit
rotations (3.39), we see that by decomposing 𝑢 𝑝𝑞 SWAP in this manner, we may apply the
SWAP gates without incurring any additional gates or depth.
Figure 6.2: Pictorial representation of the swap network used in our simulation in the case
of 𝑁 = 4 neutrinos.
At the end of a sequence of 𝑁 such combined operations we will have implemented
the full unitary in equation (6.47) while, at the same, we inverted the ordering of qubits, as
shown in Figure 6.2. This approach requires exactly the minimum number 𝑁2 of nearest-
neighbor pair operations, while the shifted ordering can be controlled completely, and
efficiently, by classical means. Note that if we were to repeat at this point the same swap
131
network in reverse order, the full unitary will correspond to a second order step for time
2𝑡 and the final ordering of qubits will be restored to its original one. This is the strategy
used in Refs. [68, 67] to study the neutrino Hamiltonian with matrix product states. In this
first implementation on quantum hardware, we focus instead on a single, linear-order, time
step.
Note that since we are only using nearest neighbor two-qubit gates, the total number
of entangling gates required for a full time evolution step is bounded from above by 3 𝑁2
while the maximum number of single qubit operations is bounded by 15 𝑁2 . As we will
see in the results presented below, the presence of a large number of arbitrary single qubit
rotations seems to be the limiting factor in implementing this scheme on the quantum
device we used in this exploration.
In order to study the build up of correlations and entanglement generated by the time-
evolution under the Hamiltonian in (6.22) we first initialize a system of 𝑁 = 4 qubits to the
following product state
|Φ0 i = |𝑒i ⊗ |𝑒i ⊗ |𝑥i ⊗ |𝑥i = |0011i (6.55)
We then preform one step of time evolution for time 𝑡 by applying the 𝑁 layers of nearest-
neighbor gates as described above. This corresponds to a single Trotter-Suzuki step for
different values of the time-step 𝑡. The four SU(2) spins representing the neutrinos are
mapped to qubits (2, 1, 3, 4) on the IBMQ Vigo quantum processor [45], whose connec-
tivity is schematically depicted in Figure 6.3. The resulting qubits are linearly connected,
allowing us to natively carry out the complete simulation scheme depicted in Figure 6.2
above.
The first observable we compute is the inversion probability 𝑃inv (𝑡), the probability that
132
Figure 6.3: Layout of the IBM Quantum Canary Processor Vigo [45]. Shown are the five
qubits, labeled from 0 to 4, and their connectivity denoted as solid black lines.
a neutrino is measured to not be in its original flavor state, of each individual neutrino as a
function of time. Note that under the simultaneous exchanges 1 ↔ 4 and 2 ↔ 3, while the
Hamiltonian in (6.22) is invariant, the flavor content of the initial state |Φ0 i gets reversed.
Therefore, in the limit of no error, 𝑃inv (𝑡) should be the same for the pairs of neutrinos
(1, 4) and (2, 3). The errors in the approximation of the propagator (6.47) do not exactly
follow this symmetry, with deviations in the range 3 − 7%. We show the results for 𝑃inv (𝑡)
obtained via the approximate evolution operator 𝑈2 (𝑡) as solid black lines in Figure 6.4,
for the pair (1, 4), and in Figure 6.5 for the pair (2, 3). The ideal, and symmetric, result is
shown instead as a purple dashed line. We see that the approximation error is very small
up to relatively large time 𝜂𝑡 ≈ 6. As discussed earlier, this is in large part an effect of
using the pair propagator 𝑈2 (𝑡) instead of the naive first order formula in equation 6.43.
The results shown in Figure 6.4 and Figure 6.5 were obtained using either the real quan-
tum device (right panels denoted QPU) or a local virtual machine simulation employing
the noise model implemented in Qiskit [3] (left panels denoted by VM) initialized with
calibration data from the device. In both plots we report the results (denoted by [bare])
obtained directly from the simulation and including only statistical errors coming from a
finite sample size (here and in the rest of the section we use 8192 repetitions, or “shots",
for every data point), as well as results obtained after performing error mitigation (denoted
133
1
VM QPU
0.8
Inversion probability Pinv
0.6
0.4
0.2 Exact result First order Trotter
Neutrino 1 [bare] Neutrino 4 [bare]
Neutrino 1 [mit] Neutrino 4 [mit]
0
0 2 4 6 8 10 0 2 4 6 8 10
−1 −1
Time t [η ] Time t [η ]
Figure 6.4: Inversion probability 𝑃inv(𝑡) for neutrinos one and four: the red circle and
brown square correspond to the bare results, the blue triangle and the green diamond are
obtained after error mitigation (see text). The left panel (VM) are virtual machine results
while the right panel (QPU) are results obtained on the Vigo [45] quantum device.
1
Exact result VM First order Trotter QPU
Neutrino 2 [bare] Neutrino 3 [bare]
0.8
Neutrino 2 [mit] Neutrino 3 [mit]
Inversion probability Pinv
0.6
0.4
0.2
0
0 2 4 6 8 10 0 2 4 6 8 10
−1 −1
Time t [η ] Time t [η ]
Figure 6.5: Inversion probability 𝑃inv (𝑡) for neutrinos two and three. The notation is the
same as for Figure 6.4.
by [mit]). This corresponds to a final post-processing step that attempts to reduce the in-
fluence of the two main sources of errors: the read-out errors associated with the imperfect
measurement apparatus and the gate error associated with the application of entangling
gates. The latter error is dealt with using a zero noise extrapolation strategy (see [31, 29]
and section 6.6 for additional details).
As seen in previous similar calculations [70, 69], the VM results obtained using the
134
0.6
fully depolarized
0.5
Inversion probability at t=0
0.4
0.3
0.2
QPU results
VM results
0.1
0
0 1 2 3 4 5 6 7 8
Noise level ε/ε0
Figure 6.6: Inversion probability 𝑃inv at the initial time 𝑡 = 0 for the first neutrino. Black
solid circles are results from the Vigo QPU [45] while the red squares correspond to
results obtained using the VM with simulated noise. Also shown are extrapolations to the
zero noise limit, for both the QPU (green line) and the VM (blue line), together with the
extrapolated value (greed triangle up and blue triangle down respectively). The dashed
orange line denotes the result for a maximally mixed state.
simulated noise are much closer to the ideal result than those obtained with the real device.
This is also reflected in the fact that the error mitigation protocol is not as successful with
the real QPU data as it is with the simulated VM data. This behavior is possibly linked to
the substantial noise caused by the presence of a large number of single qubit operations
(up to 90 degree rotations for time evolution and two for state preparation) together with
the relatively large CNOT count of 18. In fact, the performance of error mitigation for the
results with the largest state preparation circuits presented in [69] is superior to the one
obtained here, despite the use of the same device, the same error mitigation strategy and
a comparable number of entangling gates (15 CNOT in that case) while the number of
rotations was only 14. This suggests coherent errors constitute a considerable fraction of
the overall error seen in the results above.
In order to highlight the difficulties encountered when performing noise extrapolation
for this data, we plot in Figure 6.6 the results obtained from both the QPU (black circles)
135
and the VM (red squares) for the inversion probability of the first neutrino at the initial time
𝑡 = 0 together with a linear extrapolation using the first two points for the QPU (green line)
and the first three points for the VM (blue line). The exact result is of course 𝑃inv (0) = 0
and we see that neither strategy is able to predict the correct value. The horizontal dashed
line is the value expected when the system is in the maximally mixed state, corresponding
to full depolarization. As shown in the data, for the real QPU results, only the first level
of noise extrapolation contains useful information and a more gentle noise amplification
strategy, like the one proposed in [41], could provide a substantial advantage over the
strategy adopted here.
6.5 Dynamics of entanglement
In order to track the evolution of entanglement in the system we perform complete state
tomography for each of the six possible qubit pairs in our system by estimating, for each
pair (𝑘, 𝑞), the 16 expectation values
𝑘,𝑞 𝛽
𝑀𝛼,𝛽 (𝑡) = hΦ(𝑡)|𝑃 𝛼𝑘 ⊗ 𝑃𝑞 |Φ(𝑡)i, (6.56)
with 𝑃 𝛼𝑘 ∈ {𝐼, 𝑋, 𝑌 , 𝑍 } being a Pauli matrix acting on the 𝑘 th qubit and |Φ(𝑡)i the state
obtained from |Φ0 i by applying the time-evolution operator as in equation (2.13). In
principle, we may reconstruct the density matrix for the pair of qubits (𝑘, 𝑞) directly from
these expectation values as
Õ4 Õ 4
𝐷 𝑘,𝑞 𝛽
𝜌 𝑘𝑞 (𝑡) = 𝑀𝛼,𝛽 (𝑡)𝑃 𝛼𝑘 ⊗ 𝑃𝑞 . (6.57)
𝛼=1 𝛽=1
𝑘,𝑞
In practice however, we can only estimate the matrix elements 𝑀𝛼,𝛽 (𝑡) to some finite
additive precision, and the approximation in equation (6.57) is not guaranteed to be a
physical density matrix (positive definite and trace equal to 1). In this work we use the
136
common approach (see eg. [6]) of performing a maximum-likelihood (ML) optimization,
while enforcing the reconstructed density matrix 𝜌 𝑘𝑞 𝑀 𝐿 (𝑡) to be physical. We note in
passing that it is possible to devise operator bases that are more robust than the choice used
in equation (6.56) (see eg. [21]) but we don’t explore this further in our work.
𝑀 𝐿 (𝑡), we
In order to propagate the effect of statistical errors into the final estimator for 𝜌 𝑘𝑞
use a re-sampling strategy similar to what was introduced in [69] but using a Bayesian ap-
proach to determine the empirical posterior distribution. We provide a detailed description
of the adopted protocol in subsection 6.6.1.
6.5.1 Entanglement Entropies
As we mentioned in the introduction, one of the main differences between a mean field
description and the full many-body description of the dynamics of the neutrino cloud is
the absence of quantum correlations, or entanglement, in the former. Past work on the
subject [16, 71] looked at the single spin entanglement entropy defined as
𝑆 𝑘 (𝑡) = −Tr 𝜌 𝑘 (𝑡) log2 (𝜌 𝑘 (𝑡)) , (6.58)
with 𝜌 𝑘 (𝑡) the reduced density matrix of the 𝑘-th spin. A value of the entropy 𝑆 𝑘 (𝑡)
different from zero indicates the presence of entanglement between the 𝑘-th neutrino and
the rest of the system.
In our setup, we compute the one-body reduced density matrix from the maximum-
likelihood estimator of the pair density matrix defined above, explicitly
h i
𝑀𝐿 𝑀𝐿 𝑀𝐿
𝑆 𝑘;𝑞 (𝑡) = −Tr 𝜌 𝑘;𝑞 (𝑡) log2 𝜌 𝑘;𝑞 (𝑡) , (6.59)
where the reduced density matrices are computed from
h i
𝑀𝐿 𝑀𝐿
𝜌 𝑘;𝑞 (𝑡) = Tr𝑞 𝜌 𝑘𝑞 (𝑡) , (6.60)
137
1
Single spin entanglement entropy
0.8
0.6
Exact
0.4 Exact Trotter
bare QPU
3x noise QPU
Richardson
0.2 Exponential
Shifted Exponential
0
0 2 4 6 8 10
−1
Time t [η ]
Figure 6.7: Single spin entanglement entropy for neutrino 2. Black squares are bare
results obtained from the QPU, red triangles are results obtained by amplifying the noise
to 𝜖/𝜖0 = 3, the blue circles are obtained using Richardson extrapolation, the turquoise
plus symbols indicate results obtained using the standard exponential extrapolation and the
green diamonds correspond to the results obtained from a shifted exponential extrapolation
using the maximum value of the entropy (indicated as a dashed orange line).
and Tr𝑞 denotes the trace over the states of the 𝑞-th qubit. We combine the three val-
ues obtained in this way for each neutrinos as follows: the estimator for the single-spin
entanglement entropy is obtained from the average
avg 1 Õ 𝑀𝐿
𝑆 𝑘 (𝑡) = 𝑆 (𝑡), (6.61)
3 𝑞 𝑘;𝑞
summing over pairs containing the k-th spin, while as an error estimate we use the average
of the three errors.
As for the case of the inversion probability 𝑃inv (𝑡) studied in the previous section, the
substantial noise present in the QPU data prevents us from using the full set of results at
the four effective noise levels. In order to overcome this difficulty, we have performed
zero noise extrapolations using only results for effective noise levels 𝑟 = 𝜖/𝜖0 = 1, 3 and
performed a Richardson extrapolation (in this case equivalent to a simple linear fit as
done in [29]), a two point exponential extrapolation [31], and an exponential extrapolation
138
with shifted data. The latter technique consists in shifting the data for the entropy by −1
(its maximum value) so that the result, in the limit of large noise, tends to 0 instead of
log2 (2) = 1. We then shift back the result obtained after extrapolation. The exponential
extrapolation method is well suited for situations where expectation values decay to zero
as a function of the noise strength 𝜖, while maintaining a consistent sign, and this shift
allows us to make the data conform to this ideal situation (section 6.6 for more details on
the method). The impact on the efficacy of the error mitigation is dramatic as can be seen
in the results presented in Figure 6.7 for the entropy of the second neutrino (the entropies
for the other neutrinos follow a similar pattern; see subsection 6.7 for all four results).
The results with the standard exponential extrapolation are presented as the turquoise plus
symbols; they are almost the same as those obtained using Richardson extrapolation (blue
circles) and show a significant systematic error. On the contrary, the results obtained with
the shifted exponential extrapolation (green diamonds) are much closer to the expected
results with our pair propagator partition (solid black curve). We expect more general
multi-exponential extrapolation schemes, like those proposed in Refs. [37, 13], to enjoy a
similar efficiency boost in the large noise limit achieved with deep circuits.
Using the reconstructed pair density matrix 𝜌 𝑘𝑞𝑀 𝐿 (𝑡), we can clearly also directly evaluate
the entanglement entropy of the pair
h i
𝑀𝐿 𝑀𝐿 𝑀𝐿
𝑆 𝑘𝑞 (𝑡) = −Tr 𝜌 𝑘𝑞 (𝑡) log2 𝜌 𝑘𝑞 (𝑡) . (6.62)
In Figure 6.8 we show the result of this calculation for the pair (1, 2), which started as
electron flavor at 𝑡 = 0, and the pair (2, 4) which started instead as heavy flavor states.
139
2
Pair entanglement entropy
1.5
1
Exact
Exact Trotter
0.5 bare QPU
3x noise QPU
Richardson
Shifted Exponential
0
0 2 4 6 8 10 0 2 4 6 8 10
−1 −1
Time t [η ] Time t [η ]
Figure 6.8: Pair entanglement entropy for the neutrino pair (1, 2) starting as |𝑒i ⊗ |𝑒i
(left panel) and pair (2, 4) which starts as the flavor state |𝑒i ⊗ |𝑥i (right panel). Results
obtained directly from the QPU are shown as black squares (𝑟 = 1) and red triangles
(𝑟 = 3) while blue circles and green diamonds indicate mitigated results using Richardson
and the shifted exponential extrapolations respectively. For the shifted exponential ansatz
we use the maximum value of the entropy (indicated as a dashed orange line).The magenta
triangle indicates a mitigated result with shifted exponential extrapolation below zero
within errorbars.
6.5.2 Concurrence
In order to better understand these quantum correlations, we also compute the concur-
rence [88] for all the pair states. This measure of entanglement is defined for a two-qubit
density matrix as
𝐶 (𝜌) = max {0, 𝜆0 − 𝜆1 − 𝜆 2 − 𝜆 3 } , (6.63)
where 𝜆𝑖 are the square roots of the eigenvalues, in decreasing order, of the non-Hermitian
matrix
𝑀 = 𝜌 (𝑌 ⊗ 𝑌 ) 𝜌 ∗ (𝑌 ⊗ 𝑌 ) , (6.64)
with the star symbol indicating complex conjugation. The usefulness of this measure is
its relation with the entanglement of formation [43, 88], which is the minimum number of
maximally-entangled pairs needed to represent 𝜌 with an ensemble of pure states [43].
140
The definition of concurrence in equation (6.63) does not lend itself as easily to be
adapted in an error extrapolation procedure as the one we used to obtain the mitigated
results in the previous sections. This is due to the presence of the max function in the
definition of the concurrence: when the error is sufficiently strong to make the difference
in eigenvalues
e(𝜌) = 𝜆 0 − 𝜆 1 − 𝜆 2 − 𝜆 3
𝐶 (6.65)
negative, the concurrence in equation (6.63) ceases to carry information about the error
free result. For this reason, we will regard 𝐶
e as an “extended concurrence" which varies
smoothly for large error levels and perform the truncation to positive values only after the
zero noise extrapolation. The results obtained from the simulation on the Vigo QPU are
shown in Figure 6.9 for two pairs of neutrinos: pair (1, 2) starting as like spin at 𝑡 = 0 and
pair (2, 4) which started as opposite flavors. The complete set of results for all pairs can
be found in Figure 6.10 in subsection 6.7.
1.2
(1,2) (2,4)
1
Exact
0.8 Exact Trotter
bare QPU
Pair Concurrence
3X noise QPU
0.6 Richardson
Shifted Exponential
0.4
0.2
0
-0.2
-0.4
0 2 4 6 8 10 0 2 4 6 8 10
−1 −1
Time t [η ] Time t [η ]
Figure 6.9: Extended concurrence 𝐶 e for two pairs of neutrinos, (1, 2) in the left and (2, 4)
in the right panel. The convention for the curves and date point used here is the same as in
Figure 6.8. The gray area indicates the region where the concurrence 𝐶 (𝜌) is zero. The
maximum value for the concurrence is shown as a dashed orange line.
141
The bare results are shown as black squares and we can immediately notice why the
definition of 𝐶 e is so important in our case: the only bare data point with a measurable
concurrence 𝐶 (𝜌) is at 𝑡 ≈ 6.7𝜂−1 for pair (2, 4) (the right panel in Figure 6.9) while
all the other results, including those obtained with a larger noise level (red triangles), are
compatible with zero. In this situation, no mitigation of 𝐶 (𝜌) would be possible.
By keeping the negative contributions, we see that the bare results often contain a
substantial signal, while those at a higher error rate are already almost at the asymptotic
value 𝐶 e = −0.5 expected for a completely depolarized system. Note that our results seem to
e ≈ −0.44 instead of 𝐶
converge to a larger asymptotic value of 𝐶 e = −0.5. We can empirically
explain this difference as the effect of statistical fluctuations. This allowed us to perform
error extrapolation using both the Richardson and shifted exponential ansatz. Similarly
to what we observed for the entanglement entropies in the previous section, the shifted
exponential ansatz (with shift −0.5) produces consistently better results than Richardson
extrapolation. This indicates that we are more close to the asymptotic large error regime
than the small error limit used to motivate a polynomial expansion. The resilience of the
exponential extrapolations to large errors, especially augmented by an appropriate shift,
is seen here to be critical in extracting physical information from quantum simulations
carried out near the coherence limit of the device used for implementation.
6.6 Error mitigation
6.6.1 Propagation of statistical uncertainties
In this section we describe the procedure we have adopted for propagating statistical
errors in the results reported previously. We found that careful treatment of statistical
errors was important for non linear functions of the expectation values like entropy and
142
concurrence of a reconstructed density matrix.
In the following, we will symbolically denote as h𝑂i, expectation values of Pauli
operators which can be measured directly on the device. These are, for instance, the
expectation values h𝑋 𝑋i, h𝑋𝑌 i, etc. needed to reconstruct a two-qubit density matrix.
We use a Bayesian approach to perform inference from the bare counts obtained from
the device. The idea is best described initially for the simple case of a single qubit
measurement. The probability of obtaining 𝑚 measurements of the state |1i out of a total
of 𝑀 trials can be modelled as a binomial distribution
𝑀 𝑚
𝑃𝑏 (𝑚; 𝑝) = 𝑝 (1 − 𝑝) 𝑀−𝑚 , (6.66)
𝑚
with 𝑝 the probability of a |1i measurement. In order to infer the parameter 𝑝 from a given
sample 𝑚𝑖 of measurement outcomes, we use Bayes’ theorem
𝑃(𝑚𝑖 | 𝑝)𝑃( 𝑝)
𝑃( 𝑝|𝑚𝑖 ) = ∫ . (6.67)
𝑑𝑞𝑃(𝑚𝑖 |𝑞)𝑃(𝑞)
For the single qubit measurement, we use the binomial distribution as likelihood 𝑃(𝑚𝑖 | 𝑝)
and, in order to obtain a posterior 𝑃( 𝑝|𝑚𝑖 ) in closed form, we use the conjugate prior of
the binomial: the beta distribution
Γ(𝛼 + 𝛽) 𝛼−1
𝑃 𝛽 ( 𝑝; 𝛼, 𝛽) = 𝑝 (1 − 𝑝) 𝛽−1 . (6.68)
Γ(𝛼)Γ(𝛽)
Here 𝛼, 𝛽 > 0 are the parameters defining the distribution and with 𝛼 = 𝛽 = 1 we obtain a
uniform distribution. The advantage of using the Beta distribution as a prior is that, after a
measurement 𝑚𝑖 of the system is available, the parameters (𝛼0 , 𝛽0 ) of the prior distribution
get updated as
𝛼𝑖 = 𝛼0 + 𝑚𝑖 𝛽𝑖 = 𝛽0 + 𝑀 − 𝑚𝑖 . (6.69)
143
Intuitively we can interpret the parameters (𝛼0 , 𝛽0 ) of the prior as assigning an a-priori
number of measurements to the measurement outcomes, which are then updated as more
measurements are performed. In this work we used a simple uniform prior corresponding
to the choice 𝛼0 = 𝛽0 = 1 for the prior parameters.
After the inference step described above, we calculate the expectation value of a generic
non-linear function h𝐹 [𝑂]i by sampling new outcomes 𝑚0𝑘 using the posterior distribution.
More in detail, we generate a new artificial measurement 𝑚0𝑘 after the measured 𝑚𝑖 by the
following procedure
• sample a value 𝑝0𝑘 from the posterior 𝑃( 𝑝0𝑘 |𝑚𝑖 )
• sample a new measurement outcome 𝑚0𝑘 from the likelihood 𝑃𝑏 (𝑚0𝑘 ; 𝑝0𝑘 )
The new measurements 𝑚0𝑘 obtained in this way are then samples from the predictive
posterior distribution.
Using an ensemble of size 𝐿 obtained in this way, we compute h𝐹 [𝑂]i by taking an
average of the results obtained for each individual sample
𝐿
1Õ
h𝐹 [𝑂]i ≈ 𝐹 [𝑂 𝑘 ] . (6.70)
𝐿 𝑘=1
The error bars reported previously are 68% confidence intervals which we found in most
cases where well approximated by a Gaussian approximation.
This scheme is complete only for single qubit measurements but a generalization to
generic multi-qubit observables can be obtained in a straightforward way. In the situation
where we are estimating expectation values over 𝑁 qubits, the probability of measuring
a specific collection of 𝑁 bit strings 𝑚𝑖 in 𝑀 repeated trials can be described with a
multinomial distribution with 𝑁 probabilities. We use this distribution as the likelihood
144
𝑃(𝑚𝑖 | 𝑝)
® in Bayes theorem and, for similar reasons as above, we take its conjugate prior
distribution: the Dirichlet distribution (also initialized as uniform as for the Beta above).
The procedure we follow is otherwise exactly equivalent to what we described above.
6.6.2 Read-out mitigation
The qubit measurements on a real device are not perfect and it is therefore important to
understand the associated systematic errors. We refer the reader to Appendix H.1 of [69]
for a more detailed derivation of the exact procedure we employ and the motivations behind
it. Here, we instead describe the main difference with the scheme described there which
comes from the use of the Bayesian inference scheme described in the previous subsection.
In the calculations presented here, we work under the assumption that read-out errors are
independent on each qubit and perform a set of 2𝑁 calibration measurements 𝑐𝑖 (requiring
two separate executions) to extract the parameters ( 𝑒®0 , 𝑒®1 ) of the noise model (see Eq.(H1)
of [69]). In order to consistently propagate the statistical uncertainties associated from the
finite sample statistic used to estimate the noise parameters, we use an additional layer of
Bayesian sampling using a binomial prior for the two error probabilities (𝑒 0𝑛 , 𝑒 1𝑛 ) associated
to each qubit 𝑛.
Using a single pair of error probability vectors 𝜖𝑖 = ( 𝑒®0 , 𝑒®1 )𝑖 , obtained either by direct
measurement or by sampling from the posterior, we can generate a linear transformation C𝑖
that maps a set of (in general multi-qubit) measurements 𝑚𝑖 to a new set 𝑚 e𝑖 with reduced
read-out errors (see [69] for more details).
The complete procedure that we use to generate an ensemble of measurements { 𝑚 e𝑖0 } with
read-out mitigation starting from a single calibration measurement 𝑐𝑖 and Pauli operator
measurement 𝑚𝑖 is as follows
145
• sample a value 𝑝0𝑘 from the posterior 𝑃( 𝑝0𝑘 |𝑚𝑖 )
• sample a new measurement outcome 𝑚0𝑘 from the likelihood 𝑃𝑏 (𝑚𝑖0; 𝑝0𝑘 )
• for each qubit 𝑛 = {1, . . . , 𝑁 }
– sample a pair (𝑒0𝑛 0𝑛
0 , 𝑒 1 ) of error probabilities from the posterior 𝑃(𝑒 0 , 𝑒 1 |𝑐𝑖 )
𝑛 𝑛
• use the sampled error probabilities ( 𝑒®00 , 𝑒®01 ) to generate the linear transformation C0𝑘
• apply the sampled correction matrix C0𝑙 to 𝑚0𝑘 to obtain the read-out mitigated
e0𝑘
estimator 𝑚
The resulting ensemble of measurements can be used directly to estimate expectation
values and confidence intervals as described above. In this way, we avoid having to
explicitly construct the variance of the correction matrix C0𝑙 using maximum likelihood
estimation and then propagating the error perturbatively to arbitrary observables as done
in [69].
6.6.3 Zero-noise-extrapolation
For observables like inversion probability, we adopt the procedure developed in [69].
For entanglement observables we adopt a two point shifted exponential extrapolation that
we briefly describe here. We denote the entanglement observable as h𝐹 [𝑂]i (𝐿) (𝑟) where
𝐿 is the number of samples used and 𝑟 denotes the noise level of the circuit, proportional
to the number of CNOT gates in the circuit. We first note that in the case of very high
noise levels, denoted here with h𝐹 [𝑂]i(𝑟 → ∞) the density matrix corresponds to the
maximally mixed state given by 1/4. Therefore, the concurrence in this case is −1/2 and
the pair entanglement saturates to 2.
146
Using an estimate for the large noise expected value h𝐹 [𝑂]i(𝑟 → ∞), we can then
consider a simple exponential extrapolation of the form
h𝐹 [𝑂]i (𝐿) (𝑟) − h𝐹 [𝑂]i(𝑟 → ∞) = 𝐴𝐹(𝐿) 𝑒 −𝛼𝑟 , (6.71)
with 𝛼 and 𝐴𝐹(𝐿) the parameters of the model which can be obtain using results at two
different noise levels 𝑟 and 𝑟 0. The zero-noise extrapolated result in this model corresponds
to the limit 𝑟 → 0 and is given simply by the estimated 𝐴𝐹(𝐿) . More explicitly this becomes
𝑟/(𝑟−𝑟 0)
h𝐹 [𝑂]i (𝐿) (𝑟 0)
𝐴𝐹(𝐿) = h𝐹 [𝑂]i (𝐿)
(𝑟) , (6.72)
h𝐹 [𝑂]i (𝐿) (𝑟)
and the zero noise extrapolated observable is
h𝐹 [𝑂]i (𝐿) (0) = 𝐴𝐹(𝐿) + h𝐹 [𝑂]i(𝑟 → ∞) . (6.73)
Finally, the estimated statistical error is obtained by calculating the standard deviation of
the 𝐿 copies as above.
6.7 Additional data for concurrence and entanglement entropy
Here we show the full set of results for both entanglement entropy and concurrence
for all the other pairs of qubits not shown earlier. We denote with a magenta triangle,
data-points that fall below zero for the entropy as in Figure 6.9.
147
1
Single spin entropy
0.8
0.6 Exact
Exact Trotter
0.4 bare QPU
Richardson
0.2 (1) Shifted Exponential (2)
0
1
Single spin entropy
0.8
0.6
0.4
0.2 (3) (4)
0
0 2 4 6 8 10 0 2 4 6 8 10
−1 −1
Time t [η ] Time t [η ]
Figure 6.10: Single spin entanglement entropy for all four neutrinos. Black squares
are bare results obtained from the QPU, the blue circles are obtained using Richardson
extrapolation and the green diamonds correspond to the results obtained from a shifted
exponential extrapolation using the maximum value of the entropy (dashed orange line).
The magenta triangle indicates a mitigated result with shifted exponential extrapolation
below zero within errorbars.
Pair entanglement entropy
Pair entanglement entropy
2
1.5
1
0.5
(1,2) (1,3) (1,4)
0
2
1.5
1
0.5
(2,3) (2,4) (3,4)
0
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
−1 −1 −1
Time t [η ] Time t [η ] Time t [η ]
Figure 6.11: Pair entanglement entropy for all pairs of neutrinos. Black squares are bare
results obtained from the QPU, red triangles are results obtained by amplifying the noise
to 𝜖/𝜖0 = 3, the blue circles are obtained using Richardson extrapolation and the green
diamonds correspond to the results obtained from a shifted exponential extrapolation using
the maximum value of the entropy (indicated as a dashed orange line). The magenta
triangle points are mitigated results with shifted exponential extrapolation below zero
within errorbars.
148
1.2 (1,3) (1,4)
Pair Concurrence
Exact
Exact Trotter
0.8 bare QPU
Richardson
0.4
(1,2)
0
-0.4
1.2 (2,3) (2,4) (3,4)
Pair Concurrence
0.8
0.4
0
-0.4
0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10
−1 −1 −1
Time t [η ] Time t [η ] Time t [η ]
Figure 6.12: Entanglement concurrence for all the pairs of qubits. The maximum value
for the concurrence is shown as a dashed orange line.
6.8 Conclusion
In this chapter, we presented the first digital quantum simulation of the flavor dynamics
in collective neutrino oscillations using current quantum technology. The results reported
for the evolution of flavor and entanglement properties of a system with 𝑁 = 4 neutrino
amplitudes show that current quantum devices based on superconducting qubits are starting
to become a viable option for studying out-of-equilibrium dynamics of interacting many-
body systems. The reduced fidelity in the results obtained here, compared to the simulations
reported previously in [69] employing the same quantum processor and a comparable
number of entangling gates, points to the importance of controlling unitary errors associated
with the imperfect implementation of arbitrary single-qubit rotations (on average < 1% for
the device used in both works). In future work we plan to explore the use of more advanced
error mitigation strategies, such as Pauli twirling [84] or symmetry protection [81], to
achieve a better overall fidelity.
We showed the zero-noise error extrapolation using a shifted Gaussian ansatz to be
remarkably efficient in predicting the expected error-free estimator of observables. Given
149
the large circuits employed in this section, past experience with zero-noise extrapolations
(see e.g. [70, 69]) suggest the exponential ansatz to be appropriate due to the large noise
rates, and we find it to indeed outperforms Richardson extrapolation in this regime. Using
the pair concurrence together with the entropy provides a robust way to detect entanglement
even in the presence of substantial noise, like in the results shown here. We expect these
insights, and the mapping of the neutrino evolution problem into a swap network, to
prove very valuable in future explorations of out-of-equilibrium neutrino dynamics with
near-term, noisy, quantum devices.
150
CHAPTER 7
QUANTUM CIRCUIT SQUEEZING ALGORITHM
7.1 Introduction
The quantum circuit squeezing algorithm (QCSA) is a novel quantum algorithm devel-
oped in this thesis which compresses the depth of a quantum circuit, with the trade-off of
increased qubit size and required runs. The algorithm relies on two fundamental properties
of maximally entangled states: the ricochet property and entanglement swapping. QCSA
uses the combination of these two properties to replace horizontal lines of gates on a set of
qubits to a vertical line of gates on a larger set of qubits.
7.2 Maximal Entanglement
QCSA relies extensively on maximally entangled states, a state whose entanglement
entropy is maximal. A state |𝜓i is maximally entangled if the partial trace of its density
matrix 𝜌 = |𝜓i h𝜓| is equal to a multiple of the identity 𝐼; mathematically
Tr 𝐴 𝜌 𝐴𝐵 = 𝑐𝐼 𝐵 , (7.1)
where 𝑐 is a constant. The partial trace can be defined as follows: Let {|𝑎𝑖 i} and {|𝑏𝑖 i}
be the bases of Hilbert spaces 𝐻 𝐴 and 𝐻 𝐵 , respectively. Let 𝜌 𝐴𝐵 ∈ 𝐻 𝐴 ⊗ 𝐻 𝐵 be a density
matrix which can be decomposed as
Õ
𝜌 𝐴𝐵 = 𝑐𝑖 𝑗 𝑘𝑙 |𝑎𝑖 i h𝑎 𝑗 | ⊗ |𝑏 𝑘 i h𝑏 𝑙 | , (7.2)
𝑖 𝑗 𝑘𝑙
where 𝑐𝑖 𝑗 𝑘𝑙 are constants. Then the partial trace over 𝐴 is given by
Õ
Tr 𝐴 (𝜌 𝐴𝐵 ) = 𝑐𝑖 𝑗 𝑘𝑙 h𝑎 𝑚 |𝑎𝑖 i 𝑎 𝑗 𝑎 𝑚 ⊗ |𝑏 𝑘 i h𝑏 𝑙 | (7.3)
𝑖 𝑗 𝑘𝑙𝑚
151
Õ
= 𝑐𝑖 𝑗 𝑘𝑙 𝑎 𝑗 𝑎 𝑚 h𝑎 𝑚 |𝑎𝑖 i ⊗ |𝑏 𝑘 i h𝑏 𝑙 | (7.4)
𝑖 𝑗 𝑘𝑙𝑚
Õ
= 𝑐𝑖 𝑗 𝑘𝑙 𝑎 𝑗 𝑎𝑖 ⊗ |𝑏 𝑘 i h𝑏 𝑙 | . (7.5)
𝑖 𝑗 𝑘𝑙
One orthonormal set of maximally entangled states for two qubits, is the Bell state basis.
It contains the following four states
1
𝜙± = √ (|00i ± |11i) (7.6)
2
1
𝜓 ± = √ (|01i ± |10i) , (7.7)
2
which can be prepared on a quantum computer with the following short-depth circuit
|𝑞 0 i 𝐻 • (7.8)
|𝑞 1 i
where the initial state of the pair of qubits |𝑞 0 𝑞 1 i leads to the different Bell states as follows
|00i → 𝜙+ , (7.9)
|10i → |𝜙− i , (7.10)
|01i → 𝜓 + , (7.11)
|11i → |𝜓 − i . (7.12)
Each Bell state is maximally entangled because the partial trace of its density matrix is
𝐼/2. Note that preparing any of the initial states |𝑞 0 𝑞 1 i listed above does not increase the
depth of the circuit as their preparation requires at most the application of a column of two
𝑋 gates, the top of which can be combined with the 𝐻 gate to form a single-qubit gate and
the bottom of which can be executed in parallel with this new single-qubit gate. The notion
152
of a maximally entangled state can be expanded to 𝑛 qubits. One such state is what we’ll
call the generalized Bell state 𝜙+𝑛 which we’ll define to be
1 Õ
𝜙+𝑛 = √ |𝑥𝑥i , (7.13)
2𝑛 𝑥∈ℎ𝑛
where ℎ𝑛 = {𝑥 | 𝑙 (𝑥) = 𝑛} is the set of all bitstrings of length 𝑛. The states formed from
this set 𝐻𝑛 = {|𝑥i | 𝑥 ∈ ℎ𝑛 }, which we’ll call the bit-string states, are an orthonormal basis
for the Hilbert space of 𝑛-qubits. The state 𝜙+𝑛 can be prepared with the following circuit
of depth two:
|01 i 𝐻 • (7.14)
|02 i 𝐻 •
..
.
|0𝑛 i 𝐻 •
|0𝑛+1 i
|0𝑛+2 i
..
.
|02𝑛 i
where the column of 𝑛 Hadamard gates 𝐻 places the top half of the qubits in the equal
superposition of the states of 𝐻𝑛 while the ladder of CNOT gates "copies" each state of the
top half of the qubits to the state of the bottom half, maximally entangling the two sets of
qubits. The circuit is depth two because all of the CNOT gates can be run in parallel as they
each act on disjoint pairs of qubits. Note that this is simply 𝑛 copies of the quantum circuit
that prepares |𝜙+ i, (7.8) with |𝑞 0 𝑞 1 i = |00i. The proof of the maximum entanglement of
𝜙+𝑛 is given below:
©1 Õ
Tr 𝐴 𝜙+ 𝜙+ = Tr 𝐴 𝑛
|𝑥𝑥i h𝑦𝑦| ®
ª
2 𝑥,𝑦∈ℎ
« 𝑛 ¬
153
1 Õ
= h𝑥|𝑦i |𝑥i h𝑦|
2𝑛 𝑥,𝑦∈ℎ
𝑛
1 Õ
= 𝑛 𝛿𝑥𝑦 |𝑥i h𝑦|
2 𝑥,𝑦∈ℎ
𝑛
1 Õ
= 𝑛 |𝑥i h𝑥|
2 𝑥∈ℎ
𝑛
1
= 𝑛 𝐼. (7.15)
2
The maximally entangled states introduced here serve as the building blocks for QCSA.
We will now discuss the two properties of maximally entangled states that will be used to
build the algorithm.
7.3 Ricochet Property
The first property of maximally entangled states that will be used in QCSA is the
Ricochet property. It states that, for any 𝑛-qubit gate 𝐴, the following equality holds
( 𝐴 ⊗ 𝐼) 𝜙+𝑛 = (𝐼 ⊗ 𝐴𝑇 ) 𝜙+𝑛 , (7.16)
where one will recall 𝜙+𝑛 to be our previously defined (7.13) maximally entangled state
for 𝑛 qubits. The ricochet property can be proven as follows: First, write 𝐴 in terms of the
orthonormal basis of 𝑛-qubits 𝐻𝑛 (which was defined as the set of all possible bit-string
states of length 𝑛)
Õ 𝑛
𝐴= 𝑎𝑖 𝑗 |𝑖i h 𝑗 | , (7.17)
𝑖, 𝑗 ∈ℎ 𝑛
where 𝑎𝑖 𝑗 are the matrix element of 𝐴. Then, plug this rewriting of 𝐴 into the left hand
side (LHS) of the ricochet property definition (7.16)
𝑛
1 Õ
( 𝐴 ⊗ 𝐼) 𝜙+𝑛 = √ 𝑎𝑖 𝑗 (|𝑖i h 𝑗 | ⊗ 𝐼) |𝑘 𝑘i (7.18)
𝑛 𝑖𝑗𝑘
154
𝑛
1 Õ
=√ 𝑎𝑖 𝑗 |𝑖 𝑗i
𝑛 𝑖𝑗
𝑛
1 Õ
=√ 𝑎𝑖 𝑗 (𝐼 ⊗ | 𝑗i h𝑖|) |𝑘 𝑘i
𝑛 𝑖𝑗𝑘
= 𝐼 ⊗ 𝐴𝑇 𝜙+𝑛 . (7.19)
The ricochet property for single-qubit gates can be expressed via quantum circuits as:
|0i 𝐻 • 𝐴 |0i 𝐻 • (7.20)
=
|0i |0i 𝐴𝑇
The key insight here is that one can use this property to change two gates applied in series
on a single qubit to two gates applied in parallel on two qubits. This process, which we’ll
call squeezing, can be seen in terms of quantum circuits below
|0i 𝐻 • 𝐵 𝐴 |0i 𝐻 • 𝐴 , (7.21)
=
|0i |0i 𝐵𝑇
or, mathematically
( 𝐴𝐵 ⊗ 𝐼) 𝜙+𝑛 = ( 𝐴 ⊗ 𝐼) (𝐵 ⊗ 𝐼) 𝜙+𝑛
(7.22)
= ( 𝐴 ⊗ 𝐼) (𝐼 ⊗ 𝐵𝑇 ) 𝜙+𝑛
= ( 𝐴 ⊗ 𝐵𝑇 ) 𝜙+𝑛 .
155
The ricochet property for the 𝑛-qubit case, using 𝜙+𝑛 can be visualized in terms of quantum
circuits as follows
|0i 𝐻 • |0i 𝐻 • , (7.23)
|0i 𝐻 • |0i 𝐻 •
𝐴
..
.
|0i 𝐻 • |0i 𝐻 •
=
|0i |0i
|0i |0i
.. 𝐴𝑇
.
|0i |0i
and the squeezing process can additionally be visualized as
|0i 𝐻 • |0i 𝐻 • (7.24)
|0i 𝐻 • |0i 𝐻 •
𝐵 𝐴 𝐴
..
.
|0i 𝐻 • |0i 𝐻 •
=
|0i |0i
|0i |0i
.. 𝐵𝑇
.
|0i |0i
So far, we’ve only been able to squeeze (reduce) our circuit depth by a factor 2. In order to
do better, we’ll need to introduce the second property of maximally entangled states upon
which QCSA relies, entanglement swapping.
156
7.4 Entanglement swapping
The property of maximally entangled states that allows one to extend the benefits of
the ricochet property to be able to squeeze the depth of a circuit by more than half is called
entanglement swapping. To express it mathematically, let me first introduce the notation
𝜙𝑛𝑎𝑏 to mean that the generalized Bell state 𝜙+𝑛 (7.13) is applied to qubit sets 𝑎 and 𝑏;
that is
1 Õ
𝜙𝑛𝑎𝑏 = √ |𝑥 𝑎 𝑥 𝑏 i , (7.25)
2𝑛 𝑥∈ℎ𝑛
where ℎ𝑛 = {𝑥 | 𝑙 (𝑥) = 𝑛} is the set of all bitstrings of length 𝑛, while 𝑎 and 𝑏 are
sets of 𝑛-qubits. For example, for 𝑁 = 2, 𝑎 = {0, 1}, and 𝑏 = {2, 3}, we would have
ℎ𝑛 = {00, 01, 10, 11} and the generalized Bell state would be
1 Õ 1
𝜙𝑛𝑎𝑏 = √ |𝑥 {0,1} 𝑥 {2,3} i = (|0000i + |0101i + |1010i + |1111i) . (7.26)
22 𝑥∈ℎ2 2
With this notation in hand, the entanglement swapping property can then be stated as
follows:
1 𝑛
𝜙𝑛𝑏𝑐 𝜙𝑛𝑎𝑏 𝜙𝑛𝑐𝑑 = 𝜙 , (7.27)
2𝑛 𝑎𝑑
which can be proven by inserting into it the definition of the generalized Bell state (7.25),
which yields
!
1 Õ ©1 Õ
𝜙𝑛𝑏𝑐 𝜙𝑛𝑎𝑏 𝜙𝑛𝑐𝑑 = √ h𝑥 𝑏 𝑥 𝑐 | 𝑛 |𝑦 𝑎 𝑦 𝑏 𝑧 𝑐 𝑧 𝑑 i ®
ª
(7.28)
𝑛
2 𝑥∈ℎ𝑛 2
« 𝑦,𝑧∈ℎ𝑛 ¬
1 Õ
=√ |𝑦 𝑎 i h𝑥 𝑏 |𝑦 𝑏 i h𝑥 𝑐 |𝑧 𝑐 i |𝑧 𝑑 i
23𝑛/2 𝑥∈ℎ𝑛
1 Õ
=√ 𝛿𝑥𝑦 𝛿𝑥𝑧 |𝑦 𝑎 i |𝑧 𝑑 i
23𝑛/2 𝑥∈ℎ𝑛
157
!
1 1 Õ
= 𝑛 √ |𝑥 𝑎 𝑥 𝑑 i
2 2𝑛 𝑥∈ℎ𝑛
1 𝑛
= 𝜙 . (7.29)
2𝑛 𝑎𝑑
Taking the complex conjugate squared of both sides of the entanglement swapping property
(7.27)
2 1 1
𝜙𝑛𝑏𝑐 𝜙𝑛𝑎𝑏 𝜙𝑛𝑐𝑑 = 𝑛
𝜙𝑛𝑎𝑑 𝜙𝑛𝑎𝑑 = 𝑛 , (7.30)
4 4
implies that the probability of measuring qubit sets 𝑏 and 𝑐 to be in the state 𝜙+𝑛 is 1/4𝑛 .
One can test whether qubit sets 𝑏 and 𝑐 are in the state 𝜙+𝑛 by applying (𝑈𝑏𝑐 𝑛 ) † to said
qubits, measuring them, and checking if they were both measured to be in the all zero state
|0i ⊗𝑛 . Here, we’ve defined 𝑈𝑎𝑏 𝑛 as the quantum gate that takes qubit sets 𝑎 and 𝑏 from all
zero states to the generalized Bell state 𝜙𝑛𝑎𝑏 ; that is
𝑈𝑛 |0i 𝑎⊗𝑛 |0i 𝑏⊗𝑛 = 𝜙𝑛𝑎𝑏 . (7.31)
To get a better feel for the entanglement swapping property, we show here the property in
terms of quantum circuits for the 𝑛 = 1 case:
|0𝑎 i 𝐻 • |0𝑎 i 𝐻 • , (7.32)
|0𝑏 i • 𝐻
1/4
−−→
|0𝑐 i 𝐻 •
|0𝑑 i |0𝑑 i
whose mathematical description matches that of the 𝑛 = 1 case of the definition of the
property (7.27)
𝑛 † 𝑛 𝑛 1 𝑛
h00| (𝑈𝑏𝑐 ) 𝑈𝑎𝑏 𝑈𝑏𝑐 |0000i = 𝜙 (7.33)
2𝑛 𝑎𝑑
158
1 𝑛
𝜙𝑛𝑏𝑐 𝜙𝑛𝑎𝑏 𝜙𝑛𝑐𝑑 = 𝜙 , (7.34)
2𝑛 𝑎𝑑
where we’ve used the definition 7.31. Here the arrow with 1/4 above it from 7.32 implies
that qubits 𝑏 and 𝑐 were measured to be 00 with probability 1/4. Additionally, qubits 𝑏 and
𝑐 have been discarded in the circuit to the right of the arrow as they have been measured
and are therefore no longer relevant. Note that, in the 𝑛 = 1 case considered above 𝑈𝑥𝑦 𝑛
is simply the quantum circuit that prepares |𝜙+ i given in 7.8 (with 𝑥 = 𝑞 0 and 𝑦 = 𝑞 1 );
that is, a Hadamard applied to the top qubit 𝑥 and a CNOT with the qubits 𝑥 and 𝑦 being
the control and target qubits, respectively. We’ll now consider the arbitrary 𝑛 case (7.27).
Here, 𝑈𝑥𝑦𝑛 is the quantum circuit given in (7.14). This case is given in the quantum circuit
1
representation below, in which the arrow with a 4𝑛 above it implies that the qubit sets 𝑏
159
and 𝑐 were measured to be in the all zero state |0 . . . 0i:
|0i 𝐻 • |0i 𝐻 •
𝑎
|0i
𝐻 • |0i 𝐻 •
..
.
|0i
𝐻 • |0i 𝐻 •
|0i • 𝐻
𝑏
|0i
• 𝐻
..
.
|0i
• 𝐻 1
4𝑛
−−→
|0i
𝐻 •
|0i
𝑐 𝐻 •
..
.
|0i
𝐻 •
|0i |0i
|0i
|0i
𝑑 ..
.
|0i
|0i
(7.35)
7.5 Dicke Subspace Modification
The quantum circuit squeezing algorithm can be modified for circuits that preserve
Hamming weight (for example, ansatzes in second quantization for systems that preserve
particle number) so that the increase in additional measurements is reduced. To describe
this modification, let us first define what we’ll call the Dicke-Bell state |𝜙𝑛𝑘 i, which can be
160
thought of as a maximally entangled state in the sub-space of the Hilbert space for 𝑛 qubits
containing only the bit-string states that have a Hamming weight of 𝑘. Mathematically
1 Õ
𝜙𝑛𝑘 = q |𝑥𝑥i , (7.36)
𝑛 𝑛
𝑘 𝑥∈ℎ 𝑘
where ℎ𝑛𝑘 = {𝑥 | l(𝑥) = 𝑛, wt(𝑥) = 𝑘 }; that is the set of all bit-strings 𝑥 with length 𝑛 and
Hamming weight 𝑘. For example,
1
𝜙42 = √ (|1100i ⊗ |1100i + |1010i ⊗ |1010i + |1001i ⊗ |1001i (7.37)
6
+ |0110i ⊗ |0110i + |0101i ⊗ |0101i + |0011i ⊗ |0011i). (7.38)
It is analogous to the state 𝜙+𝑛 defined in (7.13) with 𝑥 being drawn from the set ℎ𝑛𝑘 instead
of ℎ𝑛 . The states created from the latter, 𝐻𝑛 = {|𝑥i |𝑥 ∈ ℎ𝑛 }, are an orthonormal basis
for the Hilbert space of 𝑛-qubits. This is as opposed to the states created from the former,
𝐻 𝑘𝑛 = {|𝑥i |𝑥 ∈ ℎ𝑛𝑘 }, which is an orthonormal basis for 𝐻 𝑘𝑛 , the subset of the Hilbert space
for 𝑛-qubits containing only quantum states formed from bit-string states with a Hamming
weight of 𝑘. The Dicke-Bell state |𝜙𝑛𝑘 i can be formed from the following circuit
|0i • (7.39)
|0i 𝑈 𝑘𝑛 •
..
.
|0i •
|0i
|0i
..
.
|0i
where 𝑈 𝑘𝑛 takes the 𝑛-qubit all-zero state to the 𝑛, 𝑘 Dicke state; that is
𝑈 𝑘𝑛 |0i ⊗𝑛 = |𝐷 𝑛𝑘 i , (7.40)
161
where the Dicke state |𝐷 𝑛𝑘 i is defined as the equal superposition of all bit-string states of
length 𝑛 and Hamming weight 𝑘; that is
1 Õ
|𝐷 𝑛𝑘 i =q |𝑥𝑥i . (7.41)
𝑛 𝑛
𝑘 𝑥∈ℎ 𝑘
The reason we’ve named the resulting state the Dicke-Bell state is now revealed; it is the
entanglement of a Dicke-state into a maximally entangled state for a constant Hamming
weight subspace. Finding a short-depth circuit decomposition for 𝑈 𝑘𝑛 (to prepare a Dicke
state) is an active area of research. It has been shown how to construct 𝑈 𝑘𝑛 with O (𝑛) depth
and O (𝑘𝑛) gates ([9] and [2]). While linear depth is certainly not as good as the constant
two depth circuits (7.14) we shall see that using the subspace modification (which requires
the preparation of Dicke states) provides a significant advantage in terms of amount of shots
(runs of quantum circuits) required. Additionally, more efficient preparation methods may
be discovered, including our novel method considered in chapter 8.
It can be shown that the ricochet property still holds for |𝜙𝑛𝑘 i. That is, for any 𝑛 by
𝑛 matrix 𝐴 𝑘 that preserves the Hamming weight (𝑘) of any state upon which it acts, the
following equality holds
( 𝐴 ⊗ 𝐼) 𝜙𝑛𝑘 = (𝐼 ⊗ 𝐴𝑇 ) 𝜙𝑛𝑘 . (7.42)
It can be proven analogously to the proof of the original ricochet property (7.18) by writing
𝐴 in terms of the orthonormal basis ℎ𝑛𝑘
Õ
𝐴𝑘 = 𝑎 𝑥𝑦 |𝑥i h𝑦| , (7.43)
𝑥,𝑦∈ℎ 𝑛𝑘
and calculating
1 Õ
( 𝐴 𝑘 ⊗ 𝐼) 𝜙𝑛𝑘 = q 𝑎 𝑥𝑦 (|𝑥i h𝑦| ⊗ 𝐼) |𝑧𝑧i (7.44)
𝑛 𝑥𝑦𝑧
𝑘
162
1 Õ
=q 𝑎 𝑥𝑦 |𝑥𝑦i
𝑛 𝑥𝑦
𝑘
1 Õ
=q 𝑎 𝑥𝑦 (𝐼 ⊗ |𝑦i h𝑥|) |𝑧𝑧i
𝑛 𝑥𝑦𝑧
𝑘
= 𝐼 ⊗ 𝐴𝑇𝑘 𝜙𝑛𝑘 ,
where 𝑥, 𝑦, 𝑧 sum over the set ℎ𝑛𝑘 . The quantum circuit representation of this ricochet
property is given below
|0i • |0i • (7.45)
|0i 𝑈 𝑘𝑛 • |0i 𝑈 𝑘𝑛 •
𝐴
..
.
|0i • |0i •
=
|0i |0i
|0i |0i
.. 𝐴𝑇
.
|0i |0i
Because 𝐴 𝑘 preserves Hamming weight, it sees the subspace 𝐻 𝑘𝑛 the same way that 𝐴
sees 𝐻 𝑛 , as the entire space that it can explore. In other words, 𝐻 𝑘𝑛 is closed under the
application of 𝐴 𝑘 just as 𝐻 𝑛 is closed under the application of 𝐴. Mathematically, ∀𝑥 ∈ 𝐻 𝑘𝑛 ,
𝐴 𝑘 𝑥 ∈ 𝐻 𝑘𝑛 just as ∀𝑥 ∈ 𝐻𝑛 , 𝐴𝑥 ∈ 𝐻𝑛 .
Entanglement swapping also holds for |𝜙𝑛𝑘 i. To explain, let me first introduce the
notation |𝜙𝑖𝑛𝑘𝑗 i to mean that the Dicke-Bell state |𝜙𝑛𝑘 i is applied to qubits 𝑖 and 𝑗; that is
E 1 Õ
𝜙𝑖𝑛𝑘𝑗 = q 𝑥𝑖 𝑥 𝑗 . (7.46)
𝑛 𝑛
𝑘 𝑥∈ℎ 𝑘
163
Then, the analogous entanglement swapping property is
1 𝑛𝑘
𝜙𝑛𝑘 𝑛𝑘 𝑛𝑘
𝑏𝑐 𝜙 𝑎𝑏 𝜙 𝑐𝑑 = 𝑛 𝜙 𝑎𝑑 , , (7.47)
𝑘
which can be proven by inserting into it the definition of the Dicke-Bell state (7.46) which
yields
© 1 Õ ª© 1 Õ
𝜙𝑛𝑘 𝑛𝑘 𝑛𝑘
= h𝑥 | |𝑦 𝑎 𝑦 𝑏 𝑧 𝑐 𝑧 𝑑 i ®
ª
𝜙 𝜙
𝑏𝑐 𝑎𝑏 𝑐𝑑
q
𝑛 𝑏 𝑥 𝑐
®
® 𝑛 (7.48)
𝑥∈ℎ 𝑛𝑘 𝑘 𝑦,𝑧∈ℎ 𝑛𝑘
« 𝑘 ¬« ¬
1 Õ
= 3/2 |𝑦 𝑎 i h𝑥 𝑏 |𝑦 𝑏 i h𝑥 𝑐 |𝑦 𝑐 i |𝑧 𝑑 i (7.49)
𝑛
𝑘 𝑥,𝑦,𝑧∈ℎ 𝑛𝑘
1 Õ
= 𝛿𝑥𝑦 𝛿𝑥𝑧 |𝑦 𝑎 i |𝑧 𝑑 i (7.50)
𝑛 3/2
𝑘 𝑥,𝑦,𝑧∈ℎ 𝑛𝑘
1 © 1 Õ ª
= 𝑛
q
𝑛 |𝑥 𝑎 i |𝑥 𝑑 i ®
® (7.51)
𝑘 𝑥∈ℎ 𝑘𝑛
« 𝑘 ¬
1 𝑛𝑘
= 𝑛 𝜙 𝑎𝑑 . (7.52)
𝑘
Taking the complex conjugate squared of both sides of the entanglement swapping property
(7.27)
2 1 1
𝜙𝑛𝑘 𝑛𝑘 𝑛𝑘
𝑏𝑐 𝜙 𝑎𝑏 𝜙 𝑐𝑑 = 2
𝜙𝑛𝑘𝑎𝑑 𝜙𝑛𝑘𝑎𝑑 = 2 , (7.53)
𝑛 𝑛
𝑘 𝑘
1
implies that the probability of measuring qubit sets 𝑏 and 𝑐 to be in the state 𝜙𝑛𝑘 is .
( 𝑛𝑘) 2
Note that the probability of success (every qubit in the qubit sets 𝑏 and 𝑐 was measured to
be zero) has gone from scaling exponentially with 𝑛 (7.27) to scaling linearly with 𝑛 (7.47),
the benefits of which will become clear soon. The entanglement swapping property can
2
be viewed in terms of quantum circuits below, in which the arrow with a 1/ 𝑛𝑘 above it
164
𝑛 2
means that the probability of equivalency is 1/ 𝑘 . The dashed line is to delineate between
the circuit between the section that entangles sets of qubits and the section that rotates the
basis as to allow one to measure in the Dicke-Bell basis.
|0i • |0i • (7.54)
|0i 𝑈 𝑘𝑛 • |0i 𝑈 𝑘𝑛 •
..
.
|0i • |0i •
|0i •
|0i • (𝑈 𝑘𝑛 ) †
..
.
|0i • 1
( 𝑛𝑘) 2
−−−→
|0i •
|0i 𝑈 𝑘𝑛 •
..
.
|0i •
|0i |0i
|0i |0i
..
.
|0i |0i
7.6 Entanglement Swapping Recursion
As we’ve seen, the procedure of entanglement swapping can be used to reduce the
length of a circuit by the order of a factor of two. However, we can further reduce the
circuit length by extending entanglement swapping in a recursive manner: First, we define
165
the base case:
† √
h0| 𝑈𝑏𝑐 𝑈𝑎𝑏 𝑈𝑐𝑑 |0i = 𝑝𝑈𝑎𝑑 |0i , (7.55)
where
𝑈𝑎𝑏 |0i = |𝜙𝑎𝑏 i , (7.56)
with |𝜙𝑎𝑏 i defined as a general maximally entangled state which is maximally entangled
in some subspace 𝐻. We define it generally here so as to cover the previously explored
cases: when 𝐻 = 𝐻 𝑛 , the full Hilbert space of 𝑛 qubits, and when 𝐻 = 𝐻 𝑘𝑛 , the subspace
of 𝐻 𝑛 restricted to states of Hamming weight 𝑘. (However, 𝐻 can be any subspace of 𝐻 𝑛 .)
The state formed by applying 𝑈𝑎𝑏 to the vacuum and the probability 𝑝 are determined by
𝐻. For example, when 𝐻 = 𝐻 𝑛 , we have 𝑈𝑎𝑏 |0i = |𝜙𝑛𝑎𝑏 i and 𝑝 = 1/4𝑛 . Meanwhile, when
𝑛 2
𝐻 = 𝐻 𝑘𝑛 , we have 𝑈𝑎𝑏 |0i = |𝜙𝑛𝑘 𝑎𝑏 i and 𝑝 = 1/ 𝑘 . This base case can be represented by
the following quantum circuit diagram.
|0i |0i (7.57)
𝑎 .. ..
. .
|0i
|0i
𝑈
|0i
𝑏 ..
.
|0i
𝑝
𝑈† →
− 𝑈
|0i
𝑐 ..
.
|0i
𝑈
|0i |0i
𝑑 .. ..
. .
|0i
|0i
166
The recursive extension of entanglement swapping can be stated mathematically as
𝑁−1
Ö Ö 𝑁 q
†
h0| 𝑈2𝑘−1,2𝑘 𝑈2𝑘−2,2𝑘−1 |0i = 𝑝 𝑁−1𝑈0,2𝑁−1 |0i , (7.58)
𝑘=1 𝑘=1
where the indices of 𝑈 refer to qubit sets. To prove this, we assume as the induction
hypothesis, that (7.58) is true and proceed to show that the statement holds when we take
𝑁 → 𝑁 + 1:
Ö 𝑁 𝑁+1
Ö
†
h0| 𝑈2𝑘−1,2𝑘 𝑈2𝑘−2,2𝑘−1 |0i ; (7.59)
𝑘=1 𝑘=1
pulling out the 𝑘 = 𝑁 term from the first product and the 𝑘 = 𝑁 and 𝑘 = 𝑁 + 1 terms from
the second product yields
𝑁−1
Ö 𝑁−1
Ö
† †
h0| 𝑈2𝑘−1,2𝑘 𝑈2𝑘−2,2𝑘−1 |0i h0| 𝑈2𝑁−1,2𝑁 𝑈2𝑁−2,2𝑁−1𝑈2𝑁,2𝑁+1 |0i ; (7.60)
𝑘=1 𝑘=1
using the base case yields
𝑁−1 𝑁−1
√ Ö
†
Ö
𝑝 h0| 𝑈2𝑘−1,2𝑘 𝑈2𝑘−2,2𝑘−1 |0i 𝑈2𝑁−2,2𝑁+1 |0i ; (7.61)
𝑘=1 𝑘=1
relabeling qubit set 2𝑁 + 1 → 2𝑁 − 1 (which is allowed as qubit set 2𝑁 − 1 has been
measured) yields
𝑁−1
Ö Ö 𝑁
†
h0| 𝑈2𝑘−1,2𝑘 𝑈2𝑘−2,2𝑘−1 |0i ; (7.62)
𝑘=1 𝑘=1
applying the induction hypothesis gives
p
𝑝 𝑁 𝑈0,2𝑁−1 |0i (7.63)
which completes the proof. The quantum circuit representation of entanglement swapping
recursion for the case 𝑁 = 3 is given below. The arrow with the 𝑝 above it implies that
167
the following circuit is equivalent to the previous with probability 𝑝. Between the first and
second circuit, the entanglement swapping property is applied amongst the first four sets
of qubits while between the second and third circuit, the entanglement swapping property
is applied amongst the remaining sets of qubits.
|0i |0i |0i (7.64)
.. .. ..
. . .
|0i |0i |0i
𝑈
|0i
..
.
|0i 𝑝
𝑈† →
− 𝑈
|0i
..
.
|0i
𝑈 𝑈
|0i |0i
.. ..
. .
|0i |0i 𝑝
𝑈† 𝑈† →
−
|0i |0i
.. ..
. .
|0i |0i
𝑈 𝑈
|0i |0i |0i
.. .. ..
. . .
|0i |0i |0i
168
7.7 The Algorithm
We will now combine everything introduced thus far (ricochet property, entanglement
swapping property, and entanglement swapping recursion) in order to construct the quantum
circuit squeezing algorithm (QCSA). Before we give a formal definition, we’ll walk through
the algorithm for single-qubit gates and a single recursion step in quantum circuit form:
Let 𝐴, 𝐵, 𝐶, 𝐷 be arbitrary single-qubit gates. Then the QCSA circuit is given by
|0i 𝐻 • 𝐴𝑇 (7.65)
|0i 𝐵 • 𝐻
|0i 𝐻 • 𝐶𝑇
|0i 𝐷
Since the qubit pairs (0, 1) and (2, 3) are each in the |𝜙+ i state, one can apply the ricochet
property to move the middle two gates (𝐵 and 𝐶 𝑇 ) outward
|0i 𝐻 • 𝐵𝑇 𝐴𝑇 (7.66)
|0i • 𝐻
|0i 𝐻 •
|0i 𝐶 𝐷
The entanglement swapping identity then tells us that, with a probability of 1/4, measuring
the middle two qubits will collapse them to the state |00i, which would result in the
following circuit
|0i 𝐻 • 𝐵𝑇 𝐴𝑇 (7.67)
|0i 𝐶 𝐷
169
where the middle two qubits have been discarded. Since the two qubits left are in the state
|𝜙+ i, the ricochet property can be applied again to move the top two gates (𝐵𝑇 and 𝐴𝑇 )
down to the bottom qubit.
|0i 𝐻 • (7.68)
|0i 𝐴 𝐵 𝐶 𝐷
One then measures the first qubit. With probability 1/2, one will measure 0 which implies
that the second qubit is in the state |0i, since the two qubits are in the entangled state |𝜙+ i.
Thus, with probability 1/2, the above circuit is equivalent to the following circuit
|0i 𝐴 𝐵 𝐶 𝐷 (7.69)
QCSA’s benefit can now be understood: If one desires to run the circuit (7.69), one can
instead run the squeezed circuit (7.65) which allows one to apply the four single-qubit gates
𝐴, 𝐵, 𝐶, 𝐷 in parallel instead of series (with the trade-off of the usage of more qubits and
the circuits only being equivalent one fourth of the time). While these two circuits have
the same depth, we will see that for a large initial depth, the depth of the squeezed circuit
can be decreased substantially through the use of QCSA. Since the final circuit (7.69) is
only equivalent to the first circuit (7.65) with probability 1/8, one must run the first circuit
eight times the number of runs one desires to run the first circuit. Assuming that these runs
cannot be done in parallel, this increases the total run time by a factor of eight. However,
we shall see later that the factor by which the run time increases need not necessarily scale
exponentially with the depth of the original circuit.
We now walk through a more robust example of QCSA. Here we will use 𝑈 and |Φi
so as to keep the description of the algorithm general. That is, |Φi can refer to either the
170
generalized Bell state 𝜙+𝑛 (7.25) or the Dicke-Bell state |𝜙𝑛𝑘 i (7.36), with 𝑈 being the
operator that transforms the all zero state into the chosen version of |Φi. The algorithm
works the same for both cases, the only difference being the probability 𝑝 of one each
subsequent circuit being equivalent after measurement. However, this probability 𝑝 will
be given for both cases at each such step as we continue. In (7.70) we start on the left hand
side (LHS) with an initial circuit which consists of six sets of qubits (𝑠0 through 𝑠5 ). These
six sets are entangled pairwise into three maximally entangled states |Φ01 Φ23 Φ45 i via the
application of 𝑈01𝑈23𝑈45 . Then we apply a column of the six gates that we actually wish to
run (𝐴0 through 𝐴5 ), transposing the even-indexed ones. Finally, we apply 𝑈 † to the inner
two pairs of qubit sets (𝑠1 ,𝑠2 ) and (𝑠3 ,𝑠4 ). To get to the right hand side (RHS) of (7.70),
we apply the ricochet property between pairs of qubit sets (𝑠0 ,𝑠1 ) and (𝑠2 ,𝑠3 ). Doing so
ricochets 𝐴1 from qubit set 𝑠1 up to qubit set 𝑠0 and ricochets 𝐴𝑇2 from qubit set 𝑠2 down
†
to qubit set 𝑠3 . This leaves space for 𝑈12 to be next to 𝑈01 and 𝑈23 , allowing us to apply
171
the entanglement swapping identity to the set of qubit sets {𝑠0 , 𝑠1 , 𝑠2 , 𝑠3 }.
|0i |0i (7.70)
𝑠0 .. 𝐴𝑇0 .. 𝐴𝑇1 𝐴𝑇0
. .
|0i
|0i
𝑈 𝑈
|0i |0i
𝑠1 .. 𝐴1 ..
. .
|0i
|0i
𝑈† 𝑈†
|0i |0i
𝑠2 .. 𝐴𝑇2 ..
. .
|0i
|0i
𝑈 = 𝑈
|0i |0i
𝑠3 .. 𝐴3 .. 𝐴2 𝐴3
. .
|0i
|0i
𝑈† 𝑈†
|0i |0i
𝑠4 .. 𝐴𝑇4 .. 𝐴𝑇4
. .
|0i
|0i
𝑈 𝑈
|0i |0i
𝑠5 .. 𝐴5 .. 𝐴5
. .
|0i
|0i
This means that the RHS of the circuit above (7.70) is equal to the LHS of the circuit below
(7.71) with probability 𝑝. The probability 𝑝 is given by 𝑝 = 1/4𝑛 if 𝑈 prepares 𝜙+𝑛 or
2
𝑝 = 1/ 𝑛𝑘 if 𝑈 prepares 𝜙𝑛𝑘 . Here 𝑛 is the number of qubits in each qubit set 𝑠𝑖 and 𝑘
is the Hamming weight that one can set as desired. To get to the LHS of (7.71) we apply
the ricochet property between pairs of qubit set (𝑠0 , 𝑠3 ) and (𝑠4 , 𝑠5 ). Doing so ricochets
𝐴2 and 𝐴3 from qubit set 𝑠3 up to qubit set 𝑠0 and ricochets 𝐴𝑇4 from qubit set 𝑠4 down to
†
qubit set 𝑠5 . This leaves space for 𝑈34 to be next to 𝑈03 and 𝑈45 , allowing us to apply the
172
entanglement swapping identity on qubit sets {𝑠0 , 𝑠3 , 𝑠4 , 𝑠5 }.
|0i |0i
𝑠0 .. 𝐴𝑇1 𝐴𝑇0 𝐴𝑇3 𝐴𝑇2 𝐴𝑇1 𝐴𝑇0
.
|0i
|0i
𝑈 𝑈
|0i |0i
𝑠3
.. =
. 𝐴2 𝐴3
|0i
|0i
𝑈† 𝑈†
|0i |0i
𝑠4 .. 𝐴𝑇4
.
|0i
|0i
𝑈 𝑈
|0i |0i
𝑠5 .. 𝐴5 𝐴4 𝐴5
.
|0i
|0i
(7.71)
This means that the RHS of the circuit above (7.71) is equal to the LHS of the circuit below
(7.72) with probability 𝑝, defined the same as before. To get to the LHS of (7.72) we apply
the ricochet property between qubit set 𝑠0 and 𝑠5 .
|0i (7.72)
𝑠0 .. 𝐴𝑇3 𝐴𝑇2 𝐴𝑇1 𝐴𝑇0
.
|0i
𝑈
|0i
𝑠5 .. 𝐴4 𝐴5
.
|0i
Applying the ricochet property between qubit sets 𝑞 0 and 𝑞 5 gives us the circuit below the
173
next circuit (7.73)
|0i (7.73)
𝑠0 ..
.
|0i
𝑈
|0i
𝑠5 .. 𝐴0 𝐴1 𝐴2 𝐴3 𝐴4 𝐴5
.
|0i
Finally, we measure the qubit set 𝑠0 . The circuit above (7.73) is equal to the circuit below
(7.74) with probability 𝑞. If 𝑈 prepares 𝜙+𝑛 , then 𝑞 = 1/2𝑛 . If 𝑈 prepares |𝜙𝑛𝑘 i then
𝑞 = 1/ 𝑛𝑘 .
|0i (7.74)
𝑠5 .. 𝐴0 𝐴1 𝐴2 𝐴3 𝐴4 𝐴5
.
|0i
Working backwards, this implies that if one desires to run the circuit above (7.74), one can
instead run the shorter depth squeezed circuit (7.74). All together, the probability 𝑝 of
5
these two circuits being equal is 𝑝 = 1/25𝑛 if 𝑈 prepares 𝜙+𝑛 or 𝑝 = 1/ 𝑛𝑘 if 𝑈 prepares
|𝜙𝑛𝑘 i.
We now describe the QCSA algorithm in full. The algorithm takes as an input, a depth
𝑛 quantum circuit consisting of 𝑚 qubits and an even integer 𝑑 between 1 and 𝑚 + 1. The
algorithm returns as an output, a depth d𝑛/𝑑e (squeezed) quantum circuit consisting of
𝑑𝑚. To begin, one groups the 𝑛 gate columns of the input circuit into 𝑛 mod 𝑑 groups of
depth d𝑛/𝑑e and 𝑑 − (𝑛 mod 𝑑) groups of depth b𝑛/𝑑c. We label these 𝑑 groups of gate
columns 𝐴0 , . . . , 𝐴𝑑−1 . With this grouping, the algorithm is described via pseudo-code in
Algorithm 7.1, below.
174
Algorithm 7.1 Quantum Circuit Squeezing Algorithm
Input: A quantum circuit consisting of 𝑚 qubits (𝑞𝑖 for 𝑖 ∈ 0, . . . , 𝑚 − 1) and 𝑑 groups
of gate columns (𝐴𝑖 for 𝑖 ∈ 0, . . . , 𝑛 − 1). An even factor 𝑑 by which the depth of the layers
of gates will be shortened.
Output: Squeezed quantum circuit consisting of 𝑑𝑚 qubits and d𝑛/𝑑e layers of gates.
success = False
while success = False do
Initialize squeezed quantum circuit with 𝑑𝑚 qubits.
for 0 ≤ 𝑖 < 𝑑/2 do
Apply 𝑈 to qubit set {𝑞 2𝑚𝑖 , . . . , 𝑞 2𝑚(𝑖+1)−1
end for
for 0 ≤ 𝑖 < 𝑑 do
if 𝑖 is even then
apply 𝐴𝑇𝑖 to the qubit set {𝑞 𝑚𝑖 , . . . , 𝑞 𝑚(𝑖+1)−1 }
end if
if 𝑖 is odd then
apply 𝐴𝑖 to the qubit set {𝑞 (𝑚(2𝑖+1) , . . . , 𝑞 𝑚(2𝑖+3)−1 }
end if
end for
for 0 ≤ 𝑖 < 𝑑/2 − 1 do
apply 𝑈 † the to qubit set {𝑞 (𝑚(2𝑖+1) , . . . , 𝑞 𝑚(2𝑖+3)−1 }
end for
for 0 ≤ 𝑖 < (𝑑 − 1)𝑚 do
Measure qubit 𝑞𝑖 to bit 𝑐𝑖
end for
if 𝑐𝑖 = 0 for all 𝑖 ∈ {0, . . . 𝑚(𝑑 − 1) − 1 then
success = True
Discard measured qubits and return circuit
end if
end while
We will now prove QCSA (Algorithm 7.1) by induction. For this proof, we define the
following parameters: let 𝑠𝑖 (for any 𝑖) denote an 𝑚-qubit set; let 𝐴𝑖 (for any 𝑖) denote a
group of gate columns, and let 𝑑 be an even positive integer. With this, QCSA for arbitrary
175
𝑑 can be stated mathematically as
𝑑−2
! 𝑑/2−2 ! 𝑑/2−1
!
Ö Ö Ö
h0𝑠 𝑘 | 𝑈𝑠†2𝑘+1 ,𝑠2𝑘+2 ( 𝐴𝑇2𝑘 ) 𝑠2𝑘
𝑘=1 𝑘=0 𝑘=0
𝑑/2−1
! 𝑑/2−1
! 𝑑−1
!
Ö Ö Ö
× ( 𝐴2𝑘+1 ) 𝑠2𝑘+1 𝑈𝑠2𝑘 ,𝑠2𝑘+1 |0𝑠 𝑘 i
𝑘=0 𝑘=0 𝑘=0
q 𝑑−1
!
Ö
= 𝑝 𝑑/2−1 ( 𝐴 𝑘 ) 𝑠2𝑑−1 𝑈𝑠0 ,𝑠2𝑑−1 0𝑠0 0𝑠 𝑑−1 . (7.75)
𝑘=0
The base case (𝑑 = 2) for QCSA is
0𝑠1 0𝑠2 𝑈𝑠†1 𝑠2 ( 𝐴𝑇0 ) 𝑠0 ( 𝐴1 ) 𝑠1 ( 𝐴𝑇2 ) 𝑠2 ( 𝐴3 ) 𝑠3 𝑈𝑠0 𝑠1 𝑈𝑠2 𝑠3 0𝑠0 0𝑠1 0𝑠2 0𝑠3 (7.76)
= 𝑝( 𝐴𝑇0 𝐴𝑇1 ) 𝑠0 ( 𝐴𝑇3 𝐴𝑇2 ) 𝑠3 𝑈𝑠0 𝑠3 0𝑠0 0𝑠3 , (7.77)
which can be proved as follows: first, we apply the ricochet property on qubit set pairs
{𝑠0 , 𝑠1 } and {𝑠2 , 𝑠3 }, which results in
𝑈𝑠†1 𝑠2 ( 𝐴𝑇0 𝐴𝑇1 ) 𝑠0 ( 𝐴𝑇3 𝐴𝑇2 ) 𝑠3 𝑈𝑠0 𝑠1 𝑈𝑠2 𝑠3 0𝑠0 0𝑠1 0𝑠2 0𝑠3 (7.78)
= ( 𝐴𝑇0 𝐴𝑇1 ) 𝑠0 ( 𝐴𝑇3 𝐴𝑇2 ) 𝑠3 𝑈𝑠†1 𝑠2 𝑈𝑠0 𝑠1 𝑈𝑠2 𝑠3 0𝑠0 0𝑠1 0𝑠2 0𝑠3 . (7.79)
Next, we apply the entanglement swapping property on the set of qubit sets {𝑠0 , 𝑠1 , 𝑠2 , 𝑠3 },
which leads to
√
𝑝( 𝐴𝑇0 𝐴𝑇1 ) 𝑠0 ( 𝐴𝑇3 𝐴𝑇2 ) 𝑠3 𝑈𝑠0 𝑠3 0𝑠0 0𝑠3 , (7.80)
where
if 𝑈 prepares |𝜙+𝑛 i
1
4𝑚 ,
𝑝= (7.81)
1
𝑚 2,
if 𝑈 prepares |𝜙𝑚𝑘 i.
(𝑘)
176
To prove QCSA, we assume as the induction hypothesis, that (7.75) is true. Taking
𝑑 → 𝑑 + 2 in (7.75) yields
𝑑
! 𝑑/2−1
! 𝑑/2
!
Ö Ö Ö
h0𝑠 𝑘 | 𝑈𝑠†2𝑘+1 ,𝑠2𝑘+2 ( 𝐴𝑇2𝑘 ) 𝑠2𝑘
𝑘=1 𝑘=0 𝑘=0
𝑑/2
! 𝑑/2
! 𝑑+1
!
Ö Ö Ö
× ( 𝐴2𝑘+1 ) 𝑠2𝑘+1 𝑈𝑠2𝑘 ,𝑠2𝑘+1 |0𝑠 𝑘 i . (7.82)
𝑘=0 𝑘=0 𝑘=0
We now pull out of their respective products: the last two 𝑈 terms 𝑈𝑠 𝑑−2 ,𝑠 𝑑−1 and 𝑈𝑠 𝑑 ,𝑠 𝑑+1 ,
the last four 𝐴 terms ( 𝐴𝑇𝑑−2 ) 𝑠 𝑑−2 , ( 𝐴𝑑−1 ) 𝑠 𝑑−1 , ( 𝐴𝑇𝑑 ) 𝑠 𝑑 , and( 𝐴𝑑+1 ) 𝑠 𝑑+1 , and the last 𝑈 † term
𝑈𝑠†𝑑−1 ,𝑠 𝑑 . This yields
𝑑
! 𝑑/2−2
! 𝑑/2−2
! 𝑑/2−2
!
Ö Ö Ö Ö
h0𝑠 𝑘 | 𝑈𝑠†2𝑘+1 ,𝑠2𝑘+2 ( 𝐴𝑇2𝑘 ) 𝑠2𝑘 ( 𝐴2𝑘+1 ) 𝑠2𝑘+1
𝑘=1 𝑘=0 𝑘=0 𝑘=0
𝑑/2−1
!"
Ö
× 𝑈𝑠2𝑘 ,𝑠2𝑘+1 𝑈𝑠†𝑑−1 ,𝑠 𝑑 ( 𝐴𝑇𝑑−2 ) 𝑠 𝑑−2 ( 𝐴𝑑−1 ) 𝑠 𝑑−1 ( 𝐴𝑇𝑑 ) 𝑠 𝑑 ( 𝐴𝑑+1 ) 𝑠 𝑑+1 (7.83)
𝑘=0
𝑑+1
!#
Ö
× 𝑈𝑠 𝑑−2 ,𝑠 𝑑−1 𝑈𝑠 𝑑 ,𝑠 𝑑+1 |0𝑠 𝑘 i . (7.84)
𝑘=0
Using the base case yields
𝑑
! 𝑑/2−2 ! 𝑑/2−2
! 𝑑/2−2
!
√ Ö Ö
†
Ö Ö
𝑝 h0𝑠 𝑘 | 𝑈𝑠2𝑘+1 ,𝑠2𝑘+2 ( 𝐴𝑇2𝑘 ) 𝑠2𝑘 ( 𝐴2𝑘+1 ) 𝑠2𝑘+1
𝑘=1 𝑘=0 𝑘=0 𝑘=0
𝑑/2−1
! Ö 𝑑+1
!
Ö
× 𝑈𝑠2𝑘 ,𝑠2𝑘+1 𝐴𝑇𝑑−2 𝐴𝑇𝑑−1 ( 𝐴𝑑+1 𝐴𝑑 ) 𝑠 𝑑+1 𝑈𝑠 𝑑−2 ,𝑠 𝑑+1 |0𝑠 𝑘 i . (7.85)
𝑠 𝑑−2
𝑘=0 𝑘=0
Absorbing the bracketed terms back into the products by relabeling 𝑠 𝑑+1 → 𝑠 𝑑 , 𝐴𝑑+1 𝐴𝑑 →
𝐴𝑑−1 , and 𝐴𝑇𝑑−2 𝐴𝑇𝑑−1 → 𝐴𝑇𝑑−2 yields
𝑑
! 𝑑/2−2
! 𝑑/2−1
!
√ Ö Ö Ö
𝑝 h0𝑠 𝑘 | 𝑈𝑠†2𝑘+1 ,𝑠2𝑘+2 ( 𝐴𝑇2𝑘 ) 𝑠2𝑘
𝑘=1 𝑘=0 𝑘=0
177
𝑑/2−1
! 𝑑/2−1
! 𝑑+1
!
Ö Ö Ö
× ( 𝐴2𝑘+1 ) 𝑠2𝑘+1 𝑈𝑠2𝑘 ,𝑠2𝑘+1 |0𝑠 𝑘 i . (7.86)
𝑘=0 𝑘=0 𝑘=0
Applying the induction hypothesis (7.75) yields
q 𝑑−1
!
Ö
𝑝 𝑑/2 ( 𝐴 𝑘 ) 𝑠2𝑑−1 𝑈𝑠0 ,𝑠2𝑑−1 |0i , (7.87)
𝑘=0
which completes the proof. The final step is to measure the the first qubit set 𝑠0 . If the
measurement succeeds (returns all zeros: 01 . . . 0𝑚 ) then the circuit is in the state
q 𝑑−1
!
Ö
𝑝 (𝑑+1)/2 ( 𝐴 𝑘 ) 𝑠2𝑑−1 𝑈𝑠0 ,𝑠2𝑑−1 |0i . (7.88)
𝑘=0
√
because the probability of success is 𝑝.
7.8 Scaling
In this section we discuss the scaling of various parameters involved in the QCSA
algorithm. Let the initial circuit that one wishes to "squeeze" have 𝑚𝑖 qubits and a depth
of 𝑛𝑖 . Let the final (squeezed) circuit have 𝑚 𝑓 qubits and a depth of 𝑛 𝑓 . Let 𝑑 be an even
integer between 1 and 𝑚 + 1 and the factor by which the initial circuit is squeezed. The
parameters of the final circuit (𝑚 𝑓 and 𝑛 𝑓 ) are given in terms of the parameters of the initial
circuit (𝑚𝑖 and 𝑛𝑖 ) as follows:
𝑚 𝑓 = 𝑑𝑚𝑖 , (7.89)
𝑛
𝑖
𝑛𝑓 = 𝑂 . (7.90)
𝑑
by the pigeon hole principle. The depth comparison equation (7.90) is written in big O
notation because the true depth of the final circuit is 𝑛𝑑𝑖 plus 2𝑑 (𝑈), twice the depth of
𝑈. If 𝑈 prepares |𝜙+𝑚 i then 𝑑 (𝑈) = 2. If 𝑈 prepares |𝜙𝑚 𝑘 i then 𝑑 (𝑈) = O (𝑚) if 𝑈 is
decomposed deterministically [9]. However, see the following chapter (8) for a potentially
178
𝑛𝑖
shorter depth decomposition. In any case, for small 𝑚𝑖 and large 𝑛𝑖 , the term 𝑑 outweighs
2𝑑 (𝑈). With these scalings (7.89 and 7.90) we can see that QCSA is most beneficial if
one wishes to run a small-qubit, long-depth circuit on a large-qubit quantum computer that
only allows short-depth circuits (and is substantially noisy). Fortunately, this is likely the
direction that many NISQ era quantum computers are going [79].
We must also consider the scaling of the time it takes to run QCSA. Recall, that QCSA
is a probabilistic algorithm, which means that it only succeeds with a certain probability.
For the full or subspace version of QCSA (𝑈 prepares |𝜙+𝑚 i or |𝜙𝑚 𝑘 i, respectively) the
probabilities of success (𝑝 full or 𝑝 sub , respectively) are
1 1
𝑝 full = 𝑚 (𝑑−1) = O 𝑚 𝑑 , (7.91)
2 𝑖 2 𝑖
!
1 1
𝑝 sub = =O . (7.92)
𝑚 𝑖 𝑑−1 𝑚𝑖𝑘 𝑑
𝑘
Note that this implies that the number of shots (runs of the quantum circuit) that are required
for the results of the squeezed circuit to match the quality of the results of the original
circuit scales inversely proportionally with 𝑝. That is, because the squeezed circuit is only
equivalent to the original circuit (when run) with probability 𝑝, one must (on average) run
the squeezed circuit 𝑝 times to equal one run of the original circuit. Thus, the number of
shots (𝑠full or 𝑠sub ) required for each version of QCSA (full or subspace, respectively) is
given by
1
𝑚𝑖 𝑑
𝑠full = =O 2 , (7.93)
𝑝 full
1
𝑘𝑑
𝑠sub = = O 𝑚𝑖 . (7.94)
𝑝 sub
The time (𝑡) it takes to run the experiment (run the quantum circuit 𝑠 times) is given by the
product of the time it takes to run the quantum circuit once (which scales as the depth 𝑛)
179
and the number of shots 𝑠. Thus, the experiment times for the initial and final circuits of
full QCSA (𝑡𝑖(full) and 𝑡 (full)
𝑓 , respectively) scale as
𝑡𝑖(full) ∝ 𝑛𝑖 , (7.95)
2𝑚 𝑖 𝑑
𝑡 (full)
𝑓 ∝ 𝑠full 𝑛 𝑓 = O 𝑛𝑖 , (7.96)
𝑑
whereas the experiment times for the initial and final circuits of subspace QCSA (section
7.5), which are given by𝑡𝑖(sub) and (𝑡 (sub) 𝑓 respectively, scale as
𝑡𝑖(sub) ∝ 𝑛𝑖 , (7.97)
!
𝑚𝑖𝑘 𝑑
𝑡 (sub)
𝑓 ∝ 𝑠sub 𝑛 𝑓 = O 𝑛𝑖 . (7.98)
𝑑
Thus, the factor by which the experiment times increase can be expressed via
𝑚𝑖 𝑑
2
𝑡𝑓(full)
=O 𝑡𝑖(full) , (7.99)
𝑑
!
𝑚 𝑘𝑑
𝑡 (sub)
𝑓 =O 𝑖
𝑡𝑖(sub) . (7.100)
𝑑
It can thus be seen that the subspace version of QCSA only takes polynomially longer to
run as the number of qubits 𝑚 increases, which is as compared to the full version of QCSA
which takes exponentially longer to run. This is the main advantage of the subspace version
of QCSA. The trade off is the longer circuit depth of 𝑈. Thus, the full version of QCSA
should only be used for a very small number of qubits 𝑚. Note that both version’s times
grow exponentially with the squeezing factor 𝑑. Thus, although it would be wonderful to
choose 𝑑 = 𝑛 (the circuit depth) and squeeze our circuit to as short a depth as possible, this
would not be advisable given the scaling of 𝑑. One potential sweet spot for 𝑑 would be as
a logarithm, that is
log2 𝑛𝑖
𝑑full = , (7.101)
𝑚𝑖
180
log𝑚𝑖 𝑛𝑖
𝑑sub = . (7.102)
𝑘
This choice of 𝑑 for full QCSA would require the initial depth 𝑛𝑖 to be exponential in the
initial number of qubits 𝑚𝑖 (that is, 𝑛𝑖 = 2𝑐𝑚𝑖 ) in order for 𝑑 = 𝑐 (with 𝑐 > 1). However,
this choice of 𝑑 for subspace QCSA would only require the initial depth 𝑛𝑖 to be polynomial
in the initial number of qubits 𝑚𝑖 (that is, 𝑛𝑖 ∝ 𝑚𝑖𝑐𝑘 ) to achieve the same. Plugging these
values of 𝑑 into 7.103 and 7.104, respectively, yields
(full) 𝑛𝑖
𝑡𝑓 =O 𝑡𝑖(full) , (7.103)
log 𝑛𝑖
2
(sub) 𝑛𝑖
𝑡𝑓 =O 𝑡 (sub) , (7.104)
log𝑚𝑖 𝑛𝑖 𝑖
which is sub-linear scaling. All of this again reiterates that QCSA is best suited for circuits
are much deeper than they are tall (number of qubits).
Finally, we discuss the qubit-connectivity required to run QCSA. Assuming that all gate
columns 𝐴 only require linear connectivity, then the full version of QCSA only requires
grid connectivity. For example, the following squeezed quantum circuit for full QCSA
with (𝑚𝑖 , 𝑛𝑖 , 𝑑) = (2, 4, 2):
|𝑞 0 i 𝐻 • (7.105)
𝐴𝑇
|𝑞 1 i 𝐻 •
|𝑞 2 i • 𝐻
𝐵
|𝑞 3 i • 𝐻
|𝑞 4 i 𝐻 •
𝐶𝑇
|𝑞 5 i 𝐻 •
|𝑞 6 i
𝐷
|𝑞 7 i
181
Figure 7.1: Graph of grid qubit-connectivity. Circled numbers represent qubits, black lines
represent that two qubits are connected, and black rectangles represent gates 𝐴.
can be run on a quantum computer with the grid architecture found on Figure 7.1, where
the black rectangles represent the two-qubit gates (𝐴𝑇 , 𝐵, 𝐶 𝑇 , and 𝐷). They inscribe the
qubits upon which their corresponding two-qubit gate acts. In the case of subspace QSCA,
as long as 𝑈 can prepare 𝜙𝑛𝑘 with linear-connectivity (reference [9] and chapter 8), then
the algorithm itself only requires linear connectivity 5.4.
7.9 Demonstration
Here we given a demonstration of QCSA. We have developed code that performs
QCSA: taking in an arbitrary quantum circuit and transforming it to the squeezed version
of itself. We tested the algorithm using the following procedure: for various initial circuit
depths (𝑛), we estimated the density matrix (𝜌est ) of the final state of each circuit (original
and squeezed) using quantum state tomography (6.56) and compared its corresponding
exact density matrix (𝜌ext ) by taking the fidelity of the two. The definition of fidelity that
we used can be defined as follows. Let 𝜌 and 𝜎 be two density matrices, where the density
matrix 𝜌 of a state |𝜙i is defined as (𝜌 = |𝜓i h𝜓|). Then their fidelity 𝐹 (𝜎, 𝜌) is defined as
182
the trace of their product
Tr(𝜎𝜌). (7.106)
Quantum state tomography is the process by which the density matrix of a 𝑛-qubit state
|𝜓i is estimated as
3
1 Õ
𝜌= 𝑆 𝛼1 ,··· ,𝛼𝑛 𝜎 (𝛼0 ) ⊗ . . . ⊗ 𝜎 (𝛼𝑛 ) , (7.107)
2𝑛 𝛼1 ,...,𝛼𝑛 =0
where 𝜎 0 = 𝐼, 𝜎 1 = 𝑋, 𝜎 2 = 𝑌 , and 𝜎 3 = 𝑍 are the Pauli operators and
𝑆 𝛼1 ,...,𝛼𝑛 = h𝜓|𝜎 (𝛼0 ) ⊗ . . . ⊗ 𝜎 (𝛼𝑛 ) |𝜓i , (7.108)
are the expectation values of the Pauli-strings 𝜎 (𝛼0 ) ⊗ . . . ⊗ 𝜎 (𝛼𝑛 ) which are estimated using
the method described in subsection 3.4.2. We studied the case with three initial qubits
(𝑚𝑖 = 3), a squeezing factor of 𝑑 = 2, and let the initial depth run over 𝑛𝑖 = 2, 3, . . . , 18.
The original circuits were populated with random three-qubit gates and each estimation of
fidelity was averaged over five runs. The number of shots used for the squeezed circuit
𝑠 𝑓 was 𝑠 𝑓 = 2𝑚(𝑑−1) 𝑠𝑖 = 8𝑠𝑖 where 𝑠𝑖 = 211 is the number of shots used for the original
circuit. The results were obtained from a noisy simulation of the quantum circuits on
a classical computer. It can be seen in Figure 7.2 that the the fidelity estimated from
the squeezed circuit is always greater than that estimated from the original circuit, the
difference becoming more pronounced as the absolute difference in their depths grows
with the increase in the original circuits depth 𝑛𝑖 . As the squeezing factor was 𝑑 = 2,
that means the ratio of the circuit depths 𝑛 𝑓 /𝑛𝑖 approaches 2 as 𝑛𝑖 grows. Here, 𝑛 𝑓 is the
squeezed circuit depth.
183
Figure 7.2: Comparison of fidelities of estimated and exact density matrices of both original
and squeezed circuits.
7.10 Conclusion
In this chapter, we have presented the quantum circuit squeezing algorithm (QCSA),
a novel quantum algorithm that trades more qubits and shots for a shorter circuit depth.
To explain the algorithm, we first introduced the concepts of maximal entanglement, the
ricochet property, and the entanglement swapping property. It was then shown how the
entanglement property can be extended recursively. Additionally, a modification of QCSA
to deal with Hamming weight preserving subspaces was introduced which ultimately
improved the time scaling qubit connectivity requirements of the algorithm. It was then
shown how all of these techniques could be combined to create QCSA. An analysis of the
scaling of various complexities of the algorithm was conducted. Finally, we demonstrated
the algorithm by running noisy simulations of it to show that the resulting squeezed circuits
were less affected by the noise due to their shorter depth. As NISQ era devices require
short depth circuits due to their high noise levels and are rapidly growing in their number
184
of qubits, QCSA has the potential to be very beneficial in the near future by allowing
researchers to run quantum circuits that otherwise would have been lost to the noise of our
era’s quantum machines.
185
CHAPTER 8
VARIATIONAL PREPARATION OF DICKE STATES
8.1 Introduction
Several chapters of this thesis involve the use of Dicke states. Chapter 4 used them
to construct the ansatz for one mapping of the problem and again used them to map the
problem to a smaller set of qubits. Additionally, chapter 5 used them as an initialization
strategy for a novel type of ansatz to solve a model in an extreme case. Finally, chapter
7 relied on the construction of them for a modification of the algorithm for Hamming
weight preserving gates. Therefore, it is clear that an efficient algorithm to construct Dicke
states would be very beneficial to many areas of the application of NISQ era algorithms to
many-body nuclear physics. While a deterministic method exists to do so that is linear in
the number of qubits ([9] and [2]) the quantum circuit constructed is often still too long to
be implemented on noisy quantum devices of the NISQ era. As an alternative we present
here a short-depth, variational algorithm with the potential to prepare Dicke states with a
shorter depth than the previously mentioned algorithm.
First, let us start with a few definitions. Recall that a Dicke state |𝐷 𝑛𝑘 i is the equal
superposition of all 𝑛-qubit states |𝑥i with Hamming weight wt(𝑥) = 𝑘; that is
− 21
𝑛 Õ
𝐷 𝑛𝑘 = |𝑥i . (8.1)
𝑘
𝑥∈{0,1} 𝑛
wt(𝑥)=𝑘
For example,
1
𝐷 42 = √ (|1100i + |1010i + |1001i + |0110i + |0101i + |0011i .) (8.2)
6
186
A Dicke-like state 𝐷𝑙 𝑘𝑛 (𝑚) is the equal superposition of 𝑚, 𝑛-qubit states |𝑥i with
Hamming weight wt(𝑥) = 𝑘 up to possible relative phases of +1,
− 12
𝑛 Õ
𝐷𝑙 𝑘𝑛 = (−1) 𝑓 (𝑥) 𝑔(𝑥) |𝑥i , (8.3)
𝑚
𝑥∈{0,1} 𝑛
wt(𝑥)=𝑘
where 𝑔(𝑥) = 0, 1 such that 𝑔(𝑥) = 1 only for 𝑚 states 𝑥. Thus a Dicke-like state is a
phased Dicke state that’s missing some bit-string states in its superposition. An example
would be
1
𝐷𝑙 24 (4) = √ (|1100i − |1010i + |1001i − |0110i .) (8.4)
6
As discussed in the motivation section, we seek to find a quantum circuit that prepares the
Dicke states with the shortest circuit depth possible. Here I propose a hybrid quantum-
classical algorithm called the Variational State Preparation Algorithm (VSPA) to find
such a circuit. VSPA is hybrid because it uses both a quantum and classical computer.
The quantum computer’s role is to implement a parameterized quantum circuit that tries to
implement a Dicke state. Upon measurement of the quantum computer, a classical computer
does post-measurement calculations to calculate a cost function which characterizes the
overlap of the prepared state with the desired Dicke state. A classical optimization algorithm
varies the parameters of the quantum circuit in order to minimize the cost function as much
as possible. This results in the quantum circuit preparing a quantum state that has a desired
overlap with the desired Dicke state.
8.2 The Algorithm
The variational quantum circuit is parameterized by a set of parameters that are tuned by
the classical optimization algorithm. The circuit consists of two parts: alternating 𝑘-state
187
preparation and variational mixing. First, the quantum computer is put into an alternating
𝑘-state which is a particular quantum state with Hamming weight 𝑘. For 𝑛 qubits, the
alternating 𝑘-state of type 𝑡 = I, II, III, | 𝐴𝑛𝑘 i 𝑡 is the quantum state whose qubits alternate
between 1 and 0 for the middle 2𝑘 qubits and are all 0 for the rest,
𝑛
𝑓 (𝑖) = 𝑖 + 1 (mod 2), − 1 − (𝑘 − 1) ≤ 𝑖 ≤ 𝑛2 − 1 + (𝑘 − 1)
2
𝐴𝑛𝑘 I
= |... 𝑓 (𝑖)...i |
𝑓 (𝑖) = 0,
otherwise
(8.5)
𝑛
𝑓 (𝑖) = 𝑖 (mod 2), − 1 − (𝑘 − 1) ≤ 𝑖 ≤ 𝑛2 − 1 + (𝑘 − 1)
2
𝐴𝑛𝑘 II
= |... 𝑓 (𝑖)...i |
𝑓 (𝑖) = 0,
otherwise
(8.6)
1 𝑛
𝐴𝑛𝑘 III
=√ 𝐴𝑘 I
+ 𝐴𝑛𝑘 II
, (8.7)
2
where 𝑘 = 1, ..., b𝑘/2c. For example
𝐴26 I
= |010100i , (8.8)
𝐴26 II
= |001010i , (8.9)
|1010i + |0101i
𝐴26 III
= |0i √ |0i . (8.10)
2
To prepare | 𝐴𝑛𝑘 i 𝑡 for 𝑘 = b𝑘/2c + 1, ..., 𝑛 one would finish by applying 𝑋 gates to all qubits
since
𝑛
𝐴𝑛−𝑘 𝑡
= 𝑋 ⊗𝑛 𝐴𝑛𝑘 𝑡
. (8.11)
For 𝑛 qubits, the alternating 𝑘-state of type I (II) can be prepared on a quantum computer
by applying an 𝑋 gates to all qubits that must be 1 (a quantum circuit of depth one). The
188
alternating 𝑘-state of type III can be prepared with a circuit of depth 3 with grid connectivity
and depth O (𝑛) with linear connectivity: Given a grid architecture, the 𝑛-qubit alternating
𝑘-state of type III can be implemented with the following constant (2) depth circuit. For
example, the circuit for 𝐴26 is shown below for six qubits:
|0i 𝐻 • • (8.12)
|0i 𝑋 •
|0i •
|0i •
|0i
|0i
√
The first two rows prepare the Bell state |𝜓 + i = (|01i + |10i)/ 2 and each subsequent pair
of CNOTs “copies” the state of each term of the Bell state and adds it onto that term. Given
a linear nearest-neighbor (LNN) architecture the 𝑛-qubit alternating 𝑘-state of type III can
be implemented with the following 2𝑛 + 1 depth circuit. For example, the circuit for 𝐴26
is shown below for six qubits:
|0i 𝐻 • • (8.13)
|0i 𝑋 × ×
|0i × • × •
|0i × ×
|0i × • ×
|0i
Once an alternating 𝑘-state has been prepared, the next step is to apply layers of
parameterized, two-qubit, Hamming weight preserving gates called partial-SWAP gates.
189
The partial-SWAP gate parameterized by 𝜃 is defined to be
©1 0 0 0ª
®
0 cos 𝜃 − sin 𝜃
®
0®
pSWAP(𝜃) = 𝑒𝑖𝜃 (𝑋𝑌 −𝑌 𝑋)/2 = ®.
® (8.14)
0 sin 𝜃 cos 𝜃 0®®
®
«0 0 0 1¬
Being in the group 𝑆𝑂 (4), the pSWAP gate can be decomposed into a circuit of depth
at most five, containing at most two CNOTs and twelve single-qubit rotations gates [4].
Note that pSWAP(𝜃) corresponds to a rotation in the subspace {|01i , |10i by the angle 𝜃
(leaving the subspace {|00i , |11i unchanged):
pSWAP(𝜃) |00i = |00i (8.15)
pSWAP(𝜃) |01i = cos 𝜃 |01i + sin 𝜃 |10i (8.16)
pSWAP(𝜃) |10i = cos 𝜃 |10i − sin 𝜃 |01i (8.17)
pSWAP(𝜃) |11i = |11i , (8.18)
thus preserving Hamming weight. Using pSWAP gates as mixing gates will always result
in a Dicke-like state since there are no complex phases in their matrix representation (only
a minus sign). A mixing layer is defined to be a single column of parameterized pSWAP
gates being applied in parallel. The variational mixing part of the VSPA algorithm consists
of several mixing layers. The first layer is applied to the middle 2𝑘 qubits. Each subsequent
layer is applied to two additional qubits than the layer before - the qubit directly above and
the qubit directly below the previous set of qubits (unless such qubit does not exist). The
pseudo-code for the variational mixing part is given below:
190
Algorithm 8.1 Variational Mixing Algorithm
Input: Number of qubits 𝑛, Hamming weight 𝑘, number of layers 𝑙, set of angles {𝜃}.
Output: Quantum circuit implementation of variational mixing section of VSPA.
index_list = [ ]
𝑖 = 𝑛2 − 1 − (𝑘 − 1)
while 𝑖 ≤ 𝑛2 − 1 + (𝑘 − 1) do
index_list.append(i)
𝑖+ = 2
end while
theta_index = 0
for 0 ≤ 𝑙 ≤ layer do
if 𝑙 ≠ 0 then new_index_list = [ ]
for 𝑖 in index_list do
new_index_list.append(𝑖 − 1)
end for
new_index_list.append(𝑖 + 1)
index_list=new_index_list
end if
for 𝑞 in index_list do
if 0 ≤ 𝑞 < 𝑛 − 1 then
Append pSWAP(𝜃[theta_index]) to qubits 𝑞 and 𝑞 +1 of the quantum circuit
theta_index + = 1
end if
end for
end for
8.3 Calculating a Cost Function
As discussed above, our use of pSWAP gates as mixing gates will always result in a
state that is the superposition of terms that all have the same Hamming weight. This should
make it easier to minimize a cost function that measures the overlap of the variationally
prepared state and a desired Dicke-like state. One good cost-function to quantify how close
191
a variationally prepared state is to an 𝑛-qubit, 𝑚-term Dicke-like state 𝐷𝑙 𝑘𝑛 (𝑚) would be:
𝑓cost (|𝜓i) = Var{|h𝑥|𝜓i| 2 |𝑥 ∈ {0, 1}𝑛 |wt(𝑥) = 𝑘 }, (8.19)
That is, the variance of the list of probabilities of measuring every possible |𝑥i with
Hamming weight 𝑘. The variance makes a good cost function as it measures the deviations
of a set from their average and the set of probabilities of an exact Dicke state would all be
exactly zero, thus having a variance of zero. The closer the probabilities are to their average,
the smaller the variance will be. These probabilities can be estimated by preparing and
measuring the VSPA circuit multiple times and dividing the number of times one measured
|𝑥i by the total number of measurements made. The variance of the coefficients’ absolute
value squared is minimized when they are all equal (equal super-position) and is a measure
of how close they are to being equal. Now, there are two variables to play with here
Í Í
that one may like to control: 𝑔𝑛 = 1 − 𝑥 𝑔(𝑥) and 𝑓𝑛 = 1 − 𝑥 𝑓 (𝑥). Minimizing the
first means maximizing the number of terms in the Dicke-like state, making the state a
closer approximation to a true Dicke state. Minimizing the second means minimizing the
number of terms that have a minus sign in front of them, again making the state a closer
approximation to a true Dicke state. Another good candidate for a cost-function would be
2
q
1 © Õ
𝑓cost (|𝜓i) = 1 − |h𝑥|𝜓i| 2 ® ,
ª
0 (8.20)
|𝑥(𝑙 )|
«𝑥∈𝑥(𝑙 0) ¬
where 𝑙 0 ≤ 𝑙, which is one minus the square of the sum of the square roots of the
probabilities. It is equal to one minus the overlap between our prepared state |𝜓i and
a desired non-phased Dicke-like state if one assumes no phases in the prepared state.
Even if the prepared state does have phases, it should still serve as a good cost-function
because it minimizes the difference between the absolute squares of the coefficients. Note
192
that 𝑓𝑛 cannot be controlled using either of these cost function alone because using the
probabilities (absolute values squared) of the coefficients erases all plus-or-minus phase
information. However, it may be discovered that having plus-or-minus phased Dicke
states is still beneficial in some circumstances such as in the multi-configuration method
(subsection 5.3.7 of chapter 5) as the solution to the pairing model for large 𝑔 is not exactly
a Dicke state but one close to it anyways.
8.4 Results
First, we created code to deterministically prepare Dicke states using the method given
in [9]. We the created code to run my novel VSPA algorithm using either of the first
two cost-functions discussed above. We have used this algorithm to prepare Dicke state
approximations to |𝐷 2𝑛 i (where 𝑛 = 4, 5, 6) that each have a higher overlap with the
true Dicke state than their deterministically ([9]) prepared counterparts when simulated
with noise (Figure 8.1). This is because, as seen in Figure 8.2, while the deterministic
preparation would in theory, create a perfect Dicke state, its depth is so much longer than
our variational circuit that, when noise is included, creates a worse approximation to the
true Dicke state. The idea is that, even though the variational method may not be able
to exactly prepare an exact Dicke state (which would require exactly minimizing the cost
function in a noisy environment), it has the potential to get closer than the deterministic
method once one accounts for the extra noise that befalls the latter. In the bottom half of
Figure 8.2, we have a histogram of the measurements obtained through noisy simulation.
The measurements that have a Hamming weight of two (the only ones which should have
been measured) are highlighted in red. All non-feasible (with a Hamming weight other
than two) are highlighted in blue. With this, it can clearly be seen that in this case, the
193
Figure 8.1: Overlap of deterministically and variationally prepared Dicke-states with true
Dicke-state.
variational method is much less affected by noise as the non-feasible states are measured
with much less frequency than in the deterministic method. Next, the number of CNOTs
is noted to be significantly less for the variational method, likely due to the large number
of the three-qubit, double controlled gates that had to be decomposed into several CNOTs
each for the deterministic method. This was also a likely factor in improving the results
as the CNOT is noisier than the single-qubit gates. Finally, we note the high number
of CNOTs in the the deterministic method circuit which connect non-neighboring qubits.
As we simulated the results on a noisy model that assumed only linear connectivity of
qubits, the compiler for the deterministic method’s circuit had to add several more swap
gates (which require 3 CNOTs) compared to the variational method’s in order to move the
non-neighboring qubits next to one another. This is because the deterministic method’s
circuit has six non-neighboring CNOTs, compared to the variational method’s two.
8.5 Conclusion
In this section, we developed the variational Dicke state preparation algorithm (VSPD),
a novel variational algorithm to prepare Dicke and Dicke-like states. We laid forth a short-
194
Figure 8.2: Comparison of deterministic and variational methods to prepare the Dicke state
|𝐷 42 i.
depth variational ansatz which can efficiently search the appropriate Hamming weight
restricted subspace. We also discussed different cost functions which could be used for the
minimization subroutine. Finally, we presented results obtained from testing the algorithm
with noisy simulations and bench-marked them favorably against a previously developed
deterministic method [9]. In the future, one might wish to compare and contrast different
ansatz and cost functions to prepare Dicke state |𝐷 𝑛𝑘 i for even more values of 𝑛 and 𝑘.
Additionally, one could look into determining a way to fix the phases 𝑓 (𝑥) of the terms of
the variational Dicke state prepared. One potential way to do this would simply to have the
ansatz and cost-function be identical to that introduced for the pairing model in chapter 5)
with the pairing strength 𝑔 set to zero. This is because, as proven in subsection 5.3.7, the
Dicke states are the eigenstates of said model. We conclude this section by reiterating the
widespread potential use of Dicke states: both as part of ansatz construction for many-body
nuclear models, and as part of the entangling process of the subspace QCSA algorithm.
195
CHAPTER 9
CONCLUSION
In this thesis, we have provided the basis for which future quantum algorithms can be
developed to solve many-body nuclear problems. For the Lipkin model, we developed
techniques to make the ansatz for different mappings of the problem lower-depth. We
also developed a mapping that cut the number of required qubits in half. For the pairing
model, we estimated the ground state energy with the variational quantum eigensolver
(VQE) and bench-marked the results against the classical algorithm pair coupled cluster
theory (pCCD). We also developed techniques to reduce the circuit depth for the unitary pair
coupled cluster doubles (UpCCD) ansatz for quantum computers with circular connectivity
that has a lower depth than the ansatz for devices with linear connectivity (5.4). We also
developed an algorithm for finding the energies of excited states for the model. Finally, we
developed a novel ansatz in which one starts the quantum computer in a Dicke state in order
to solve the pairing model for large values constant pairing strength 𝑔. In the collective
neutrino oscillations chapter, we presented the first digital simulation of the model on a
real quantum computer. We used this to study the time-evolution and entanglement of the
system. We applied error mitigation techniques to improve the accuracy of our results.
Then we presented the quantum circuit squeezing algorithm and showed how it could be
used to reduce the circuit depth and therefore improve the results of arbitrary quantum
circuits. Finally, we presented our novel algorithm to variationally prepared Dicke states.
As this work was meant to serve as a springboard of which future developments in the
field could be accomplished, we discuss here several future extensions and applications of
this work. For the Lipkin model, one could explicitly compare the different mappings and
196
ansatzes discussed to have better evidence that the one we choose to pursue is truly the
best choice. One could do this by running tests on the gradients of the wave-function to
see if either of the methods lends itself more easily to the problem of the barren plateau
(exponentially vanishing gradients). In terms of the pairing model, one would desire to
compare our iterative quantum excited states algorithm to other algorithms that attempt
to accomplish the same thing. Additionally, one would like to run the VQE algorithms
of this section on the quantum computers of the near future (via the cloud) that would
allow for the frequent calls to the quantum computer that is required for VQE. As for the
collective neutrino oscillation simulations, one would like to compare various permutations
of the Trotter terms that make up the ansatz to see if putting the terms with the highest
commutator with the Hamiltonian first would improve the results on an actual quantum
computer. The reasoning behind this is that, because the qubits decohere exponentially,
putting the most important terms of the ansatz first would give said terms the least noisy
effect on the quantum state. For the quantum circuit squeezing algorithm, one could apply
it to a pre-existing algorithm like VQE to see if it improves the accuracy of the results.
Finally, the variational preparation of Dicke states could similarly be applied to pre-existing
algorithms that would benefit from the initialization of their circuits into a Dicke state. In
the far future, one would like to see the improved quantum computers of that time be
used to tackle the problems of nuclear physics that are too complex for today’s classical
computers, serving as a light in the deep hole of our ignorance, shone towards the hidden
treasures buried deep below.
197
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APPENDIX A
SU(2) COMMUTATION RELATIONS
In this appendix we prove the mapping between the angular momentum SU(2) operators
(𝐽+ , 𝐽− , and 𝐽𝑧 ) and the fermionic creation and annihilation operators (𝑎 and 𝑎 † ). The
mappings (4.10) and (4.11) are shown to hold through the demonstration that the SU(2)
commutation relations (4.14) and (4.15) hold under the substitution of fermionic operators
for SU(2) operators from the mappings. Here we define 𝑗 𝑚± = 𝑎 †𝑚± 𝑎 𝑚∓ . We have that
Õ Õ
[𝐽+ , 𝐽− ] = [ 𝑗 𝑚+ , 𝑗 𝑛− ]
𝑚 𝑛
Õ
= [ 𝑗 𝑚+ , 𝑎 †𝑛− 𝑎 𝑛+ ]
𝑚𝑛
Õ
= [ 𝑗 𝑚+ , 𝑎 †𝑛− ] 𝑎 𝑛+ + 𝑎 †𝑛− [ 𝑗 𝑚+ , 𝑎 𝑛+ ]
𝑚𝑛
Õ
= 𝛿𝑚𝑛 𝑎 †𝑚+ 𝑎 𝑛+ − 𝑎 †𝑛− 𝑎 𝑚−
𝑚𝑛
Õ
= 𝑎 †𝑚+ 𝑎 𝑚+ − 𝑎 †𝑚− 𝑎 𝑚−
𝑚
= 2𝐽𝑧 , (A.1)
and
1Õ † Õ
[𝐽𝑧 , 𝐽± ] = [ 𝜎𝑎 𝑚𝜎 𝑎 𝑚𝜎 , 𝑗 𝑛± ]
2 𝑚𝜎 𝑛
1Õ
= 𝜎[𝑎 †𝑚𝜎 𝑎 𝑚𝜎 , 𝑗 𝑛± ]
2 𝑚𝑛𝜎
1Õ †
†
= 𝜎 𝑎 𝑚𝜎 [𝑎 𝑚𝜎 , 𝑗 𝑛± ] + [𝑎 𝑚𝜎 , 𝑗 𝑛± ] 𝑎 𝑚𝜎
2 𝑚𝑛𝜎
1Õ † †
= 𝜎 𝛿𝜎± 𝑎 𝑚𝜎 𝑎 𝑛∓ − 𝛿−𝜎± 𝑎 𝑛± 𝑎 𝑚𝜎
2 𝑚𝑛𝜎
207
Õ
=± 𝑎 †𝑚± 𝑎 𝑚∓
𝑚
= ±𝐽± , (A.2)
where we’ve used the commutations of the SU(2) ladder operator and the fermionic oper-
ators which are derived below
𝑗 𝑝𝜎 , 𝑎 †𝑞𝜏 = 𝑎 †𝑝𝜎 𝑎 𝑝−𝜎 , 𝑎 †𝑞𝜏
= 𝑎 †𝑝𝜎 𝑎 𝑝−𝜎 , 𝑎 †𝑞𝜏 + 𝑎 †𝑝𝜎 , 𝑎 †𝑞𝜏 𝑎 𝑝−𝜎
= 𝑎 †𝑝𝜎 𝑎 𝑝−𝜎 , 𝑎 †𝑞𝜏 − 2𝑎 †𝑞𝜏 𝑎 𝑝−𝜎
+ 𝑎 †𝑝𝜎 , 𝑎 †𝑞𝜏 − 2𝑎 †𝑞𝜏 𝑎 †𝑝𝜎 𝑎 𝑝−𝜎
= 𝛿 𝑝𝑞 𝛿−𝜎𝜏 𝑎 †𝑝𝜎 − 2 𝑎 †𝑝𝜎 , 𝑎 †𝑞𝜏 𝑎 𝑝−𝜎
= 𝛿 𝑝𝑞 𝛿−𝜎𝜏 𝑎 †𝑝𝜎 (A.3)
†
𝑗 𝑝𝜎 , 𝑎 𝑞𝜏 = − 𝑎 𝑞𝜏 , 𝑘 𝑝𝜎 = − 𝑘 †𝑝𝜎 , 𝑎 †𝑞𝜏
†
= − 𝑘 𝑝−𝜎 , 𝑎 †𝑞𝜏 = −(𝛿 𝑝𝑞 𝛿𝜎𝜏 𝑎 †𝑝−𝜎 ) †
= −𝛿 𝑝𝑞 𝛿𝜎𝜏 𝑎 𝑝−𝜎 . (A.4)
208
APPENDIX B
NORMALIZED HARTREE-FOCK ANSATZ
In this appendix we derive the normalization of the Hartree-Fock ansatz. The definition of
a coherent state, applied to the group SU(2), gives that the SU(2) coherent state takes the
form
|𝜏i = 𝑒 𝜁 𝐽+ −𝜁 𝐽− |0i ,
¯
(B.1)
where 𝜉 = (𝜃/2)𝑒 −𝑖𝜙 . We wish to write the coherent state in the form
|𝜏i0 = 𝑁𝑒 𝜏𝐽+ |0i , (B.2)
which we will do through the use of the SU(2) generating function
h𝜏| 𝑒 𝜏 𝑒 𝛼0 𝐽0 𝑒 𝛼− 𝐽− |𝜏i (B.3)
h i 2𝐽
= (1 + |𝜏| 2 ) −2𝐽 𝑒 𝛼0 /2 |𝜏| 2 + 𝑒 −𝛼0 /2 (𝛼+ 𝜏 ∗ + 1)(𝛼− 𝜏 + 1) , (B.4)
and the BCH equation
2
𝑒 𝜁 𝐽+ −𝜁 𝐽− = 𝑒 𝜏𝐽+ 𝑒 ln(1+|𝜏| )𝐽0 −𝜏𝐽
¯ ¯ −
𝑒 . (B.5)
If the two forms of the coherent state are equal, then
1 (B.6)
0
= h𝜏|𝜏i (B.7)
= 𝑁 h𝜏| 𝑒 𝜏𝐽+ |0i (B.8)
= 𝑁 h𝜏| 𝑒 𝜏𝐽+ 𝑒 −(𝜁 𝐽+ −𝜁 𝐽− ) |𝜏i
¯
(B.9)
2
)𝐽0 𝜏𝐽
= 𝑁 h𝜏| 𝑒 ln(1+|𝜏| 𝑒 ¯ −
|𝜏i (B.10)
209
h 2 2
i 2𝐽
= (1 + |𝜏| 2 ) −2𝐽 𝑒 ln(1+|𝜏| )/2 |𝜏| 2 + 𝑒 −ln(1+|𝜏| )/2 (1 + |𝜏| 2 ) (B.11)
h 1 1
i 2𝐽
= (1 + |𝜏| 2 ) −2𝐽 (1 + |𝜏| 2 ) 2 |𝜏| 2 + (1 + |𝜏| 2 ) − 2 (1 + |𝜏| 2 ) (B.12)
= (1 + |𝜏| 2 ) 𝐽 , (B.13)
which implies that 𝑁 = (1 + |𝜏| 2 ) −𝐽 .
210
APPENDIX C
LIPKIN HAMILTONIAN FOR HARTREE-FOCK
In this appendix, we derive the matrix element h𝜏|𝐻|𝜎i where 𝐻 is the Lipkin Hamiltonian
and
Ω
|𝜏i = (1 + |𝜏| 2 ) − 2 𝑒 𝜏𝐽+ |0i , (C.1)
Ω
|𝜎i = (1 + |𝜎| 2 ) − 2 𝑒 𝜎𝐽+ |0i , (C.2)
are two arbitrary 𝑆𝑈 (2) coherent states. Here |0i = |𝐽 − 𝐽i, 𝜏 = tan(𝜃/2)𝑒 −𝑖𝜙 , 𝜎 =
0
tan(𝜃 0/2)𝑒 −𝑖𝜙 , and Ω = 𝐽/2. Expanding
Ω
h𝜏| 𝐻 |𝜎i = (1 + |𝜎|) − 2 h𝜏| 𝐻𝑒 𝜎𝐽+ |0i (C.3)
Ω Ω
= (1 + |𝜏|) 2 (1 + |𝜎|) − 2 h𝜏| 𝐻𝑒 (𝜎−𝜏)𝐽+ |𝜏i (C.4)
Ω Ω
= (1 + |𝜏|) 2 (1 + |𝜎|) − 2 (C.5)
(𝜎−𝜏)𝐽+ 1 2 2 (𝜎−𝜏)𝐽+
× 𝜖 h𝜏| 𝐽0 𝑒 |𝜏i − 𝑉 h𝜏| (𝐽+ + 𝐽− )𝑒 |𝜏i . (C.6)
2
Using the SU(2) generating function
h𝜏| 𝑒 𝛼− 𝐽− 𝑒 𝛼0 𝐽0 𝑒 𝛼+ 𝐽+ |𝜏i (C.7)
h iΩ
= (1 + |𝜏| 2 ) −Ω 𝑒 −𝛼0 /2 + 𝑒 𝛼0 /2 ( 𝜏¯ + 𝛼− )(𝜏 + 𝛼+ ) , (C.8)
we calculate each term separately, starting with
h𝜏| 𝐽0 𝑒 (𝜎−𝜏)𝐽+ |𝜎i
𝜕
= h𝜏| 𝑒 𝛼0 𝐽0 𝑒 (𝜎−𝜏)𝐽+ |𝜏i
𝜕𝛼0 𝛼0 =0
1 𝜕 h −𝛼0 /2 iΩ
𝛼0 /2
= 𝑒 + ¯
𝜏𝜎𝑒
(1 + |𝜏| 2 ) Ω 𝜕𝛼0 𝛼0 =0
211
Ω/2
= ¯ 𝛼0 /2 − 𝑒 −𝛼0 /2 )(𝜎 𝜏𝑒
(𝜎 𝜏𝑒 ¯ 𝛼0 /2 + 𝑒 −𝛼0 /2 ) Ω−1
(1 + |𝜏| 2 ) Ω 𝛼0 =0
Ω/2
= (𝜎 𝜏¯ − 1)(𝜎 𝜏¯ + 1) Ω−1 , (C.9)
(1 + |𝜏| 2 ) Ω
followed by
h𝜏| 𝐽+2 𝑒 (𝜎−𝜏)𝐽+ |𝜎i
𝜕2
= 2
h𝜏| 𝑒 (𝛼+ +𝜎−𝜏)𝐽+ |𝜏i
𝜕𝛼+ 𝛼+ =0
1 𝜕 2
= 2 Ω 𝜕𝛼 2
¯ + + 𝜎)] Ω
[1 + 𝜏(𝛼
(1 + |𝜏| ) + 𝛼+ =0
Ω(Ω − 1) 2
= 2 Ω
¯ + + 𝜎)] Ω−2
( 𝜏¯ ) [1 + 𝜏(𝛼
(1 + |𝜏| ) 𝛼+ =0
Ω(Ω − 1) 2
= 2 Ω
¯ Ω−2 ,
( 𝜏¯ )(1 + 𝜏𝜎) (C.10)
(1 + |𝜏| )
followed by
h𝜏| 𝐽−2 𝑒 (𝜎−𝜏)𝐽+ |𝜎i
𝜕2
= 2
h𝜏| 𝑒 𝛼− 𝐽− 𝑒 (𝜎−𝜏)𝐽+ |𝜏i
𝜕𝛼− 𝛼− =0
1 𝜕2
= [1 + ( 𝜏¯ + 𝛼− )𝜎] Ω
(1 + |𝜏| 2 ) Ω 𝜕𝛼−2 𝛼− =0
Ω(Ω − 1)
= (𝜎 2 ) [1 + ( 𝜏¯ + 𝛼− )𝜎] Ω−2
(1 + |𝜏| 2 ) Ω 𝛼− =0
Ω(Ω − 1)
= ¯ Ω−2 .
(𝜎 2 )(1 + 𝜏𝜎) (C.11)
(1 + |𝜏| 2 ) Ω
Combining yields
Ω
h𝜏| 𝐻 |𝜎i = (Ω/2) [(1 + |𝜏| 2 )(1 + |𝜎| 2 )] − 2
h i
× 𝜖 (𝜎 𝜏¯ − 1)(𝜎 𝜏¯ + 1) Ω−1 − 𝑉 (Ω − 1) 𝜏¯ 2 + 𝜎 2 (1 + 𝜏𝜎) ¯ Ω−2 . (C.12)
212
APPENDIX D
DICKE STATE LEMMAS
In this appendix, we inductively prove the following two lemmas involving Dicke states
Õ𝑛 p
𝐴†𝑝 𝐷 𝑛𝑘 = (𝑛 − 𝑘)(𝑘 + 1) 𝐷 𝑛𝑘+1 , (D.1)
𝑝=1
Õ 𝑛 p
𝐴 𝐷 𝑛𝑘 = (𝑛 − 𝑘 + 1)(𝑘) 𝐷 𝑛𝑘−1 , (D.2)
𝑝=1
The base case for the first lemma (D.1) is given by the case (𝑛, 𝑘) = (1, 0) below
𝐴1† 𝐷 10 = 𝐴1† |01 i
= |11 i
= 𝐷 11
p
= (1 − 0)(0 + 1) 𝐷 11 . (D.3)
Similarly, the base case for the second lemma (D.2) is given by the case (𝑛, 𝑘) = (1, 1)
below
𝐴1 𝐷 11 = 𝐴1 |11 i
= |01 i
= 𝐷 10
p
= (1 − 1 + 1)(1) 𝐷 11 . (D.4)
We prove the first lemma first by assuming, as our induction hypothesis, that (D.1) is true
and show that it still holds when we take 𝑛 → 𝑛 + 1, as follows
𝑛+1 𝑛
r r !
Õ Õ
† ª 𝑘 𝑛 − 𝑘 + 1
𝐴†𝑝 𝐷 𝑛+1 = 𝐴†𝑝 + 𝐴𝑛+1 𝐷 𝑛𝑘−1 |1i + 𝐷 𝑛𝑘 |0i ,
©
(D.5)
+ +
𝑘 ®
𝑝=1
𝑛 1 𝑛 1
« 𝑝=1 ¬
213
where we took out the 𝑛 + 1 term from the sum and used the recursive definition of the
Dicke state (5.117). Using the first base case (D.3) yields
𝑛
r r ! r
©Õ † ª 𝑘 𝑛 − 𝑘 + 1 𝑛−𝑘 +1 𝑛
𝐴𝑝® 𝐷 𝑛𝑘−1 |1i + 𝐷 𝑛𝑘 |0i + 𝐷 𝑘 |1i . (D.6)
𝑛 + 1 𝑛 + 1 𝑛 + 1
« 𝑝=1 ¬
Applying the induction hypothesis (D.1) and then combining the first and last terms yields
r r
𝑛−𝑘 +1 𝑛 p 𝑛−𝑘 +1 𝑛
(𝑘 + 1) 𝐷 𝑘−1 |1i + (𝑛 − 𝑘)(𝑘 + 1) 𝐷 |0i (D.7)
𝑛+1 𝑛+1 ! 𝑘
r r
p 𝑘 +1 𝑛 𝑛−𝑘 𝑛
= (𝑛 − 𝑘 + 1)(𝑘 + 1) 𝐷 𝑘−1 |1i + 𝐷 |0i (D.8)
𝑛+1 𝑛+1 𝑘
p
= (𝑛 − 𝑘 + 1)(𝑘 + 1) 𝐷 𝑛+1 𝑘+1 , (D.9)
which completes the proof.
We now prove the second lemma first by assuming, as our induction hypothesis, that
(D.2) is true and show that it still holds when we take 𝑛 → 𝑛 + 1, as follows
𝑛+1 𝑛
r r !
Õ
𝑛+1 ©Õ 𝑘 𝑛 𝑛−𝑘 +1 𝑛
= 𝐴 𝑝 + 𝐴𝑛+1 ® |1i + 𝐷 𝑘 |0i , (D.10)
ª
𝐴𝑝 𝐷 𝑘 𝐷
𝑝=1 𝑝=1
𝑛 + 1 𝑘−1 𝑛+1
« ¬
where we took out the 𝑛 + 1 term from the sum and used the recursive definition of the
Dicke state (5.117). Using the second base case (D.4) yields
𝑛
r r ! r
©Õ ª 𝑘 𝑛 − 𝑘 + 1 𝑘
𝐴𝑝® 𝐷 𝑛𝑘−1 |1i + 𝐷 𝑛𝑘 |0i + 𝐷 𝑛𝑘−1 |0i (D.11)
𝑝=1
𝑛 + 1 𝑛 + 1 𝑛 + 1
« ¬
Applying the induction hypothesis (D.2) and then combining the first and last terms yields
r r
p 𝑘 −1 𝑛 𝑘
𝑘 (𝑛 − 𝑘 + 2) 𝐷 𝑘−2 |1i + (𝑛 − 𝑘 + 2) 𝐷𝑛 |0i (D.12)
𝑛+1 𝑛 + 1 𝑘−1!
r r
p 𝑘 −1 𝑛 𝑛−𝑘 +2 𝑛
= 𝑘 (𝑛 − 𝑘 + 2) 𝐷 𝑘−2 |1i + 𝐷 𝑘−1 |0i (D.13)
𝑛+1 𝑛+1
p
= 𝑘 (𝑛 − 𝑘 + 2) 𝐷 𝑛+1 𝑘−1 , (D.14)
which completes the proof.
214
APPENDIX E
STATE-OVERLAP ALGORITHM
In this appendix we explain the efficient state-overlap algorithm that is used to find the
eigenenergies of excited states. Consider two states |𝜓i = 𝑎 |0i+𝑏 |1i and |𝜙i = 𝑐 |0i+𝑑 |1i.
The overlap between these two states is
|h𝜓|𝜙i| 2 = (𝑎 ∗ 𝑐 + 𝑏 ∗ 𝑑) (𝑎𝑐∗ + 𝑏𝑑 ∗ ) (E.1)
= |𝑎| 2 |𝑐| 2 + |𝑏| 2 |𝑑| 2 + 𝑎 ∗ 𝑏𝑐𝑑 ∗ + 𝑎𝑏 ∗ 𝑐∗ 𝑑. (E.2)
Now consider the circuit
|𝜓i • 𝐻 (E.3)
|𝜙i
The progression of the state through the above circuit is
|𝜓𝜙i = 𝑎𝑐 |00i + 𝑎𝑑 |01i + 𝑏𝑐 |10i + 𝑏𝑑 |11i (E.4)
→ 𝑎𝑐 |00i + 𝑎𝑑 |01i + 𝑏𝑐 |11i + 𝑏𝑑 |10i (E.5)
1
→ √ (𝑎𝑐 + 𝑏𝑑) |00i + (𝑎𝑑 + 𝑏𝑐) |01i (E.6)
2
+ (𝑎𝑐 − 𝑏𝑑) |10i + (𝑎𝑑 − 𝑏𝑐) |11i . (E.7)
The probability that |𝜓𝜙i = |11i is
1
𝑃11 = (𝑎𝑑 − 𝑏𝑐)(𝑎𝑑 − 𝑏𝑐) ∗ (E.8)
2
1 2 2
|𝑎| |𝑑| + |𝑏| 2 |𝑐| 2 − 𝑎𝑏 ∗ 𝑐∗ 𝑑 − 𝑎 ∗ 𝑏𝑐𝑑 ∗ .
= (E.9)
2
215
Note that
𝑃00 + 𝑃01 + 𝑃10 − 𝑃11 (E.10)
=1 − 2𝑃11 (E.11)
=1 − |𝑎| 2 |𝑑| 2 + |𝑏| 2 |𝑐| 2 − 𝑎𝑏 ∗ 𝑐∗ 𝑑 − 𝑎 ∗ 𝑏𝑐𝑑 ∗
(E.12)
=|𝑎| 2 |𝑐| 2 + |𝑏| 2 |𝑑| 2 + 𝑎𝑏 ∗ 𝑐∗ 𝑑 + 𝑎 ∗ 𝑏𝑐𝑑 ∗ (E.13)
=|h𝜓|𝜙i| 2 , (E.14)
where we’ve used the fact that 1 = |𝑎| 2 + |𝑏| 2 . This can be generalized to
• ··· 𝐻 , (E.15)
• ··· 𝐻
|𝜓i
.. ..
. .
··· •
𝐻
···
···
|𝜙i
.. ..
. .
···
as explained in [36].
216
APPENDIX F
PAIR COMMUTATION RELATIONS
In this appendix, we use the fermionic anti-commutation relations (2.39) to prove the pair
fermionic commutation relations (5.6-5.8). The first commutation relation is proved as
follows
[ 𝐴 𝑝 , 𝐴†𝑞 ] = [𝑎 𝑝− 𝑎 𝑝+ , 𝑎 †𝑞+ 𝑎 †𝑞− ] (F.1)
= 𝑎 𝑝− 𝑎 †𝑞+ [𝑎 𝑝+ , 𝑎 †𝑞− ] + 𝑎 𝑝− [𝑎 𝑝+ , 𝑎 †𝑞+ ] 𝑎 †𝑞−
+ 𝑎 †𝑞+ [𝑎 𝑝− , 𝑎 †𝑞− ] 𝑎 𝑝+ + [𝑎 𝑝− , 𝑎 †𝑞+ ] 𝑎 †𝑞− 𝑎 𝑝+ (F.2)
= 𝑎 𝑝− 𝑎 †𝑞+ , 𝑎†𝑞− } − 2𝑎 𝑝− 𝑎 †𝑞+ 𝑎 †𝑞− 𝑎 𝑝+
{𝑎 𝑝+
+ 𝑎 𝑝− {𝑎 𝑝+ , 𝑎 †𝑞+ } 𝑎 †𝑞− − 2𝑎 𝑝− 𝑎 †𝑞+ 𝑎 𝑝+ 𝑎 †𝑞−
+ 𝑎 †𝑞+ {𝑎 𝑝− , 𝑎 †𝑞− } 𝑎 𝑝+ − 2𝑎 †𝑞+ 𝑎 †𝑞− 𝑎 𝑝− 𝑎 𝑝+
, 𝑎†𝑞+ } 𝑎 †𝑞− 𝑎 𝑝+ − 2𝑎 †𝑞+ 𝑎 𝑝− 𝑎 †𝑞− 𝑎 𝑝+
+{𝑎 𝑝− (F.3)
= 𝛿 𝑝𝑞 (𝑎 𝑝− 𝑎 †𝑞− + 𝑎 †𝑞+ 𝑎 𝑝+ )
− 2𝑎 𝑝− 𝑎 †𝑞+ † † †
{𝑎𝑞− , 𝑎 𝑝+ } − 2𝑎 𝑞+ {𝑎 𝑞− , 𝑎 𝑝− } 𝑎 𝑝+ (F.4)
= 𝛿 𝑝𝑞 (𝑎 𝑝− 𝑎 †𝑞− − 𝑎 †𝑞+ 𝑎 𝑝+ ) (F.5)
† †
𝑎 𝑝− 𝑎 𝑝− − 𝑎 𝑝+ 𝑎 𝑝+ if 𝑝 = 𝑞
= , (F.6)
0
if 𝑝 ≠ 𝑞
which we now compare to
𝛿 𝑝𝑞 (1 − 𝑁 𝑝 ) = 𝛿 𝑝𝑞 (𝛿 𝑝−𝑝− − 𝑎 †𝑝+ 𝑎 𝑝+ − 𝑎 †𝑝− 𝑎 𝑝− ) (F.7)
= 𝛿 𝑝𝑞 (𝑎 †𝑝− 𝑎 𝑝− + 𝑎 𝑝− 𝑎 †𝑝− − 𝑎 †𝑝+ 𝑎 𝑝+ − 𝑎 †𝑝− 𝑎 𝑝− ) (F.8)
= 𝛿 𝑝𝑞 (𝑎 𝑝− 𝑎 †𝑝− − 𝑎 †𝑝+ 𝑎 𝑝+ ) (F.9)
217
† †
𝑎 𝑝− 𝑎 𝑝− − 𝑎 𝑝+ 𝑎 𝑝+ if 𝑝 = 𝑞
= , (F.10)
0
if 𝑝 ≠ 𝑞
and note that they match, thus proving the first pair anti-commutation relation (5.6). The
second pair anti-commutation relation is proved as follows
Õ
[𝑁 𝑝 , 𝐴†𝑞 ] = [ 𝑎 †𝑝𝜎 𝑎 𝑝𝜎 , 𝑎 †𝑞+ 𝑎 †𝑞− ] (F.11)
𝜎
Õ
= [𝑎 †𝑝𝜎 𝑎 𝑝𝜎 , 𝑎 †𝑞+ 𝑎 †𝑞− ] (F.12)
𝜎
Õ
= 𝑎 †𝑝𝜎 𝑎 †𝑞+ [𝑎 𝑝𝜎 , 𝑎 †𝑞− ] + 𝑎 †𝑝𝜎 [𝑎 𝑝𝜎 , 𝑎 †𝑞+ ] 𝑎 †𝑞−
𝜎
+ 𝑎 †𝑞+ [𝑎 †𝑝𝜎 , 𝑎 †𝑞− ] 𝑎 𝑝𝜎 + [𝑎 †𝑝𝜎 , 𝑎 †𝑞+ ] 𝑎 †𝑞− 𝑎 𝑝𝜎 (F.13)
Õ
= (𝑎 †𝑝𝜎 𝑎 †𝑞+ {𝑎 𝑝𝜎 , 𝑎 †𝑞− } − 2𝑎 †𝑝𝜎 𝑎 †𝑞+ 𝑎 †𝑞− 𝑎 𝑝𝜎
𝜎
+ 𝑎 †𝑝𝜎 {𝑎 𝑝𝜎 , 𝑎 †𝑞+ } 𝑎 †𝑞− − 2𝑎 †𝑝𝜎 𝑎 †𝑞+ 𝑎 𝑝𝜎 𝑎 †𝑞−
+ 𝑎 †𝑞+ {𝑎 †𝑝𝜎 , 𝑎 †𝑞− } 𝑎 𝑝𝜎 − 2𝑎 †𝑞+ 𝑎 †𝑞− 𝑎 †𝑝𝜎 𝑎 𝑝𝜎
+ {𝑎 †𝑝𝜎 , 𝑎 †𝑞+ } 𝑎 †𝑞− 𝑎 𝑝𝜎 − 2𝑎 †𝑞+ 𝑎 †𝑝𝜎 𝑎 †𝑞− 𝑎 𝑝𝜎 ) (F.14)
Õ
= [𝛿 𝑝𝑞 (𝛿𝜎+ 𝑎 †𝑝𝜎 𝑎 †𝑞− + 𝛿𝜎− 𝑎 †𝑝𝜎 𝑎 †𝑞+ )
𝜎
− 2(𝑎 †𝑝𝜎 𝑎 †𝑞+ {𝑎 †𝑞− , 𝑎 𝑝𝜎 } + 𝑎 †𝑞+ {𝑎 †𝑞− , 𝑎 †𝑝𝜎 } 𝑎 𝑝𝜎 )] (F.15)
Õ
= 𝛿 𝑝𝑞 [𝛿𝜎+ 𝑎 †𝑝𝜎 𝑎 †𝑞− + 𝛿𝜎− 𝑎 †𝑝𝜎 𝑎 †𝑞+ − 2𝛿𝜎− 𝑎 †𝑝𝜎 𝑎 †𝑞+ ] (F.16)
𝜎
= 𝛿 𝑝𝑞 (𝑎 †𝑝+ 𝑎 †𝑞− + 𝑎 †𝑝− 𝑎 †𝑞+ − 2𝑎 †𝑝− 𝑎 †𝑞+ ) (F.17)
= 𝛿 𝑝𝑞 (𝑎 †𝑝+ 𝑎 †𝑞− − 𝑎 †𝑝− 𝑎 †𝑞+ ) (F.18)
† †
if 𝑝 = 𝑞
2𝑎 𝑝+ 𝑎 𝑝−
= , (F.19)
0
if 𝑝 ≠ 𝑞
218
since {𝑎 †𝑝+ , 𝑎 †𝑝− } = 0, which we now compare to
2𝛿 𝑝𝑞 𝐴†𝑝 = 2𝛿 𝑝𝑞 𝑎 †𝑝+ 𝑎 †𝑝− (F.20)
† †
if 𝑝 = 𝑞
2𝑎 𝑝+ 𝑎 𝑝−
= , (F.21)
0
if 𝑝 ≠ 𝑞
and note that they match, thus proving the second pair anti-commutation relation (5.7).
The third anti-commutation relation (5.8) can be derived from the second (5.7) as follows
†
𝑁 𝑝 , 𝐴𝑞 = −[𝑁 𝑝 , 𝐴†𝑞 ]
(F.22)
= −2𝛿 𝑝𝑞 𝐴†𝑝 , (F.23)
since 𝑁 𝑝† = 𝑁 𝑝 .
219