Advances in the programmability of quantum many-body systems have stimulated the development of novel computational and scientific applications. In this dissertation, we address a few fundamental and practical questions on using programmable quantum systems for analog quantum simulations and quantum optimization algorithms. We also design concrete experimental schemes to engineer the many-body dynamics that can be harnessed for these and other applications. On analog quantum simulations, we initiate the rigorous study of the physical resource complexity required to simulate a quantum Hamiltonian by another whose underlying interaction graph is simpler. In particular, we show a surprising result that unlike the classical setting, reducing the graph degree to a constant is impossible in general unless the interaction energy diverges with system size n. Instead, we develop a new construction where such degree-reduction becomes possible using O(poly(n)) interaction energy, which is exponentially better than what was known previously. On quantum optimization, we investigate the properties of a general-purpose Quantum Approximate Optimization Algorithm (QAOA). We develop a parameter-optimization procedure for the QAOA that is exponentially more efficient than standard strategies and reveal a mechanism of the algorithm to exploit non-adiabatic operations. We also analyze the typical-case performance of the QAOA on the Sherrington-Kirkpatrick spin glass problem and find that it can outperform the classical semi-definite programming algorithm. Lastly, we develop efficient and robust methods to experimentally implement many-body dynamics for algorithmic and physics applications. We design several experimental implementations based on neutral atoms with Rydberg interactions, which can be applied to quantum optimization algorithms, preparation of entangled many-body states, and engineering novel interactions between photons.
Advances in the programmability of quantum many-body systems have stimulated the development of novel computational and scientific applications. In this dissertation, we address a few fundamental and practical questions on using programmable quantum systems for analog quantum simulations and quantum optimization algorithms. We also design concrete experimental schemes to engineer the many-body dynamics that can be harnessed for these and other applications. On analog quantum simulations, we initiate the rigorous study of the physical resource complexity required to simulate a quantum Hamiltonian by another whose underlying interaction graph is simpler. In particular, we show a surprising result that unlike the classical setting, reducing the graph degree to a constant is impossible in general unless the interaction energy diverges with system size n. Instead, we develop a new construction where such degree-reduction becomes possible using O(poly(n)) interaction energy, which is exponentially better than what was known previously. On quantum optimization, we investigate the properties of a general-purpose Quantum Approximate Optimization Algorithm (QAOA). We develop a parameter-optimization procedure for the QAOA that is exponentially more efficient than standard strategies and reveal a mechanism of the algorithm to exploit non-adiabatic operations. We also analyze the typical-case performance of the QAOA on the Sherrington-Kirkpatrick spin glass problem and find that it can outperform the classical semi-definite programming algorithm. Lastly, we develop efficient and robust methods to experimentally implement many-body dynamics for algorithmic and physics applications. We design several experimental implementations based on neutral atoms with Rydberg interactions, which can be applied to quantum optimization algorithms, preparation of entangled many-body states, and engineering novel interactions between photons.
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.
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.
Long-range interactions are the source of many equilibrium and out-of-equilibrium quantum many-body phenomena. Analog simulators based on ionic, atomic, superconducting, and molecular systems provide a natural platform to obtain these interactions using vibration- and photon-mediated processes. Recent experimental advances, such as their integration in multi-mode cavities and waveguides, or the use of Raman-assisted transitions, enable dynamical control over both the strength and the spatial range of these interactions, thereby rendering them programmable. Here, we develop a hybrid classical-quantum toolbox that exploits this tunability to enhance many-body state preparation in analog simulators beyond fixed-connectivity architectures. Our approach is based on classical pre-compilation in homogeneous small systems, whose optimized parameters are extrapolated iteratively to larger system sizes, and then refined on the quantum hardware using noise-aware hybrid re-optimization and error-mitigation techniques. We benchmark this strategy across several fermionic, spin-1/2, and spin-1 models, demonstrating orders-of-magnitude improvements in fidelity and energy estimation for system sizes ranging from 100 to 1000 particles. Finally, we show that the combination of such high-fidelity programmable state preparation techniques with tunable-range out-of-equilibrium dynamics enables controlled studies of many-body thermalization in regimes accessible to current experimental platforms. Our results establish programmable long-range interactions as a powerful resource for next-generation analog quantum simulators.
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.
Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we consider the application of quantum computers to scientific computing and combinatorial optimization. We study five problems. The first three deal with quantum algorithms for computational problems in science and engineering, including quantum simulation of physical systems. In particular, we study quantum algorithms for numerical computation, for the approximation of ground and excited state energies of the Schrödinger equation, and for Hamiltonian simulation with applications to physics and chemistry. The remaining two deal with quantum algorithms for approximate optimization. We study the performance of the quantum approximate optimization algorithm (QAOA), and show a generalization of QAOA, the quantumalternatingoperatoransatz, particularly suitable for constrained optimization problems and low-resource implementations on near-term quantum devices.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.