The simulation of quantum systems is the most promising candidate for computational advantagesof large scale, fault-tolerant quantum computers over classical computers. Two fundamental prob- lems that appear when simulating quantum systems on quantum computers are the preparation of good input states to the simulation and the problem of decomposing the time evolution operator ???? into primitive gates that are available on the quantum computer. This thesis presents two new algorithms for solving these problems. The first is a time evolution algorithm that approximates the time evolution operator by combining two previously developed techniques. We bound the error of this composite approach and determine analytic conditions on when this can lead to cost savings of the overall simulation. These savings are then verified through extensive numerics. This simulation algorithm can then be used as a subroutine to prepare thermal states, which are typical input states that have a high overlap with the ground state. Our algorithm is inspired by Hamiltonian Monte Carlo, which is a classical sampling algorithm that utilizes Hamiltonian dynamics to sample from arbitrary probability distributions. We bound the cost of our quantum thermal state preparation algorithm in terms of the total amount of time needed to simulate the Hamiltonian of the system in four scenarios: single qubit systems, truncated harmonic oscillators, non-degenerate systems where one knows the eigenvalues, and non-degenerate systems where no eigenvalue knowledge is present. We numerically demonstrate that this algorithm works as guaranteed and can significantly outperform analytic worst-case bounds.
The simulation of quantum systems is the most promising candidate for computational advantagesof large scale, fault-tolerant quantum computers over classical computers. Two fundamental prob- lems that appear when simulating quantum systems on quantum computers are the preparation of good input states to the simulation and the problem of decomposing the time evolution operator ???? into primitive gates that are available on the quantum computer. This thesis presents two new algorithms for solving these problems. The first is a time evolution algorithm that approximates the time evolution operator by combining two previously developed techniques. We bound the error of this composite approach and determine analytic conditions on when this can lead to cost savings of the overall simulation. These savings are then verified through extensive numerics. This simulation algorithm can then be used as a subroutine to prepare thermal states, which are typical input states that have a high overlap with the ground state. Our algorithm is inspired by Hamiltonian Monte Carlo, which is a classical sampling algorithm that utilizes Hamiltonian dynamics to sample from arbitrary probability distributions. We bound the cost of our quantum thermal state preparation algorithm in terms of the total amount of time needed to simulate the Hamiltonian of the system in four scenarios: single qubit systems, truncated harmonic oscillators, non-degenerate systems where one knows the eigenvalues, and non-degenerate systems where no eigenvalue knowledge is present. We numerically demonstrate that this algorithm works as guaranteed and can significantly outperform analytic worst-case bounds.
In this thesis, we present three innovations that can be used to simulate quantum dynamics on both quantum and classical computers. Our primary innovation is an efficient quantum algorithm for simulating time-dependent Hamiltonian evolution of general input states on a quantum computer. Given conditions on the smoothness of the Hamiltonian, the complexity of the algorithm is close to linear in the evolution time, and therefore is comparable to algorithms for time-independent Hamiltonians. In addition, we show how the complexity can be reduced by optimizing the time steps. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian. In contrast to previous work, which allowed an oracle query to yield an arbitrary number of bits or qubits, we assign a cost for each bit or qubit accessed, revealing hitherto unnoticed simulation costs. We also account for discretization errors in the time and the representation of the Hamiltonian. This work is enabled by our development of a decomposition scheme based on Lie-Trotter- Suzuki product formulae for ordered operator exponentials. We show, by counterexample, that Lie-Trotter-Suzuki approximations may be lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. We address this problem by presenting a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds for the approximation error. We also provide a new analysis of the evolution of quantum adiabatic systems in this thesis. Our work contains the tightest upper bounds that are currently known, and the first tight lower bounds, for the error in the adiabatic approximation applied to fixed Hamiltonians in the limit of slow evolution. We use our results to address the Marzlin- Sanders counterexample Hamiltonian. Specifically, we provide sufficiency conditions for the validity of the adiabatic approximation and demonstrate that the Marzlin-Sanders counterexample fails to satisfy them. We also find a new class of time-dependent Hamiltonians that are related to the Marzlin0Sanders counterexample Hamiltonian, but nonetheless obey the adiabatic approximation for sufficiently slow evolutions.
The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations.
Quantum computers are expected to significantly outperform classical computers for certain computational tasks. One of the most promising applications is the simulation of many-body systems such as complex molecules or materials. In this thesis, we present several quantum algorithms for simulating many-body systems that can provide considerably better scaling than prior approaches. First, we describe a fully coherent quantum algorithm for simulating molecular dynamics within the Born-Oppenheimer approximation that leads to super-polynomially better scaling with respect to the error tolerance than previous work. Our algorithm utilizes the Koopman-von Neumann formulation of classical mechanics. This allows us to express the classical equations of motion as unitary dynamics which can be efficiently simulated on a quantum computer. We demonstrate that these simulations can be performed either at constant energy or at constant temperature. The latter allows us to estimate thermodynamic properties such as the Helmholtz free energy of the system. Next, we present two closely related quantum algorithms for solving continuous optimization problems such as finding the equilibrium configurations of molecules. These algorithms do not require any explicit gradient estimations. Instead, we encode the optimization problem into the dynamics of a quantum or classical system with friction. The dynamics are governed by a carefully chosen time-dependent Hamiltonian and thus can be simulated using efficient Hamiltonian simulation algorithms. We provide evidence that our fully coherent quantum algorithms can, in certain cases, provide exponentially better asymptotic scaling than approaches based on gradient descent. Lastly, we present a quantum algorithm for preparing the ground state of a set of interacting systems that can circumvent the exponential scaling suggested by the Van Vleck catastrophe. Our algorithm utilizes a divide-and-conquer strategy that repeatedly uses phase estimation to merge ground states of increasingly larger subsystems. We provide rigorous bounds for this approach and show that, under certain assumptions, our algorithm scales only quasi-polynomially with the number of systems.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
Simulating the Hamiltonian dynamics of quantum systems is one of the most promising applications of digital quantum computers. In this dissertation, we develop an understanding of quantum simulation algorithms concerning their design, analysis, implementation, and application. We implement three leading simulation algorithms, employing diverse techniques to tighten their error analyses and optimize circuit implementations. We produce concrete resource estimates for simulating a Heisenberg spin system, a problem arising in condensed matter physics that is otherwise difficult to solve on a classical computer. The resulting circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry, suggesting the simulation of spin systems as a promising candidate for an early demonstration of practical quantum computation. We design new simulation algorithms by using classical randomness. We show that by simply randomizing how the terms in the Hamiltonian are ordered, one can prove stronger bounds for product formulas and thereby give more efficient quantum simulations. We also develop a classical sampler for time-dependent Hamiltonians, using which we give a simulation algorithm that substantially improves over previous approaches when the Hamiltonian varies significantly with time. We propose a general theory to analyzing product formulas, an approach to quantum simulation widely used in experimental demonstrations but whose error scaling was poorly understood. Our approach directly exploits the commutativity of Hamiltonian, overcoming the limitations of prior error analyses. We prove new speedups of product formulas for simulating many quantum systems, including simulations of nearest-neighbor lattice systems, second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, and clustered Hamiltonians, nearly matching or even outperforming the best previous results in quantum simulation. We accompany our analysis with numerical calculation, which suggests that the bounds also have nearly tight constant prefactors. We identify applications of quantum simulation to designing other quantum algorithms and improving quantum Monte Carlo methods. We develop an algorithmic framework ``quantum singular value transformation'' using techniques from quantum simulation and apply it to implement principal component regression. We also apply our new analysis of product formulas and obtain improved quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets.