Digital quantum Hamiltonian simulation is, by now, a relatively mature field of study; however, new investigations are justified by the importance of quantum simulation for scientific and societal applications. In this dissertation, we discuss several advances in circuit-based Hamiltonian simulation.First, following two introductory chapters, we consider the mitigation of Trotter errors using Chebyshev interpolation, a standard yet powerful function approximation technique. Implications for estimating time-evolved expectation values are discussed, and a rigorous analysis of errors and complexity show near optimal estimation of dynamical expectation values using only Trotter and constant overhead. We supplement our theoretical findings with numerical demonstrations on a 1D random Heisenberg model.Next, we introduce a computational reduction from time dependent to time independent Hamiltonian simulation based on the standard (?, ?′) technique. Our approach achieves two advances. First, we provide an algorithm for simulating time dependent Hamiltonians using qubitization, an optimal algorithm that cannot handle time-ordering directly. Second, we provide an algorithm for time dependent simulation using a natural generalization of multiproduct formulas, achieving higher accuracies than product formulas while retaining commutator scaling. Rigorous performance analyses are performed for both algorithms, and simple numerics demonstrate the effectiveness of the multiproduct formulas procedure at reducing Trotter error.Finally, we consider several practical methods for near-term quantum simulation. First, we consider the analog quantum simulation of bound systems with discrete scale invariance using trapped-ion systems, with applications to Efimov physics. Next, we discuss the Projected Cooling Algorithm, a method for preparing bound states of non-relativistic quantum systems with localized interactions based on the dispersion of unbound states. Lastly, we discuss the Rodeo Algorithm, a probabilistic, iterative, phase-estimation-like protocol which is resource-frugal and effective at measuring and preparing eigenstates. Concluding remarks and possible future directions of research are given in a brief final chapter.
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