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We show a divide and conquer approach for simulating quantum mechanical systems on quantum computers. We can obtain fast simulation algorithms using Hamiltonian structure. 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 very mild assumptions.

The next few years will be exciting as prototype universal quantum processors emerge, enabling the implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation and which have the potential to significantly expand the breadth of applications for which quantum computers have an established advantage. A leading candidate is Farhi et al.’s quantum approximate optimization algorithm, which alternates between applying a cost function based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the quantum alternating operator ansatz, is the consideration of general parameterized families of unitaries rather than only those corresponding to the time evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach, in the spirit of the quantum approximate optimization algorithm, to a wide variety of approximate optimization, exact optimization, and sampling problems. In addition to introducing the quantum alternating operator ansatz, we lay out design criteria for mixing operators, detail mappings for eight problems, and provide a compendium with brief descriptions of mappings for a diverse array of problems.

Quantum computers have demonstrable ability to solve problems at a scale beyond brute-force classical simulation. Interest in quantum algorithms has developed in many areas, particularly in relation to mathematical optimization — a broad field with links to computer science and physics. In this Review, we aim to give an overview of quantum optimization. Provably exact, provably approximate and heuristic settings are first explained using computational complexity theory, and we highlight where quantum advantage is possible in each context. Then, we outline the core building blocks for quantum optimization algorithms, define prominent problem classes and identify key open questions that should be addressed to advance the field. We underscore the importance of benchmarking by proposing clear metrics alongside suitable optimization problems, for appropriate comparisons with classical optimization techniques, and discuss next steps to accelerate progress towards quantum advantage in optimization.This Review discusses quantum optimization, focusing on the potential of exact, approximate and heuristic methods, core algorithmic building blocks, problem classes and benchmarking metrics. The challenges for quantum optimization are considered, and next steps are suggested for progress towards achieving quantum advantage.

Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli \\(Z\\) operators (Ising spin operators) with the terms of the sum corresponding to the function's Fourier expansion. For many classes of functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses. We give composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks. We apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results. A primary goal of this paper is to provide a \\(\\textit{design toolkit for quantum optimization}\\) which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to demystify the various constructions appearing in the literature.

Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for fixed parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry. We provide several new ZX-diagram rewrite rules and generalizations for this setting. In particular, we give formal rules for dealing with linear combinations of ZX-diagrams, where the relative complex-valued scale factors of each diagram must be kept track of, in contrast to most previously studied single-diagram realizations where these coefficients can be effectively ignored. This allows us to directly import a number useful relations from the operator analysis to ZX-calculus setting, including causal cone and quantum gate commutation rules. We demonstrate that the diagrammatic approach offers useful insights into algorithm structure and performance by considering several ansatze from the literature including realizations of hardware-efficient ansatze and QAOA. We find that by employing a diagrammatic representation, calculations across different ansatze can become more intuitive and potentially easier to approach systematically than by alternative means. Finally, we outline how diagrammatic approaches may aid in the design and study of new and more effective quantum circuit ansatze.

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.

Parameterized quantum circuits are attractive candidates for potential quantum advantage in the near term and beyond. At the same time, as quantum computing hardware not only continues to improve but also begins to incorporate new features such as mid-circuit measurement and adaptive control, opportunities arise for innovative algorithmic paradigms. In this work we focus on measurement-based quantum computing protocols for approximate optimization, in particular related to quantum alternating operator ans\"atze (QAOA), a popular quantum circuit model approach to combinatorial optimization. For the construction and analysis of our measurement-based protocols we demonstrate that diagrammatic approaches, specifically ZX-calculus and its extensions, are effective for adapting such algorithms to the measurement-based setting. In particular we derive measurement patterns for applying QAOA to the broad and important class of QUBO problems. We further outline how for constrained optimization, hard problem constraints may be directly incorporated into our protocol to guarantee the feasibility of the solution found and avoid the need for dealing with penalties. Finally we discuss the resource requirements and tradeoffs of our approach to that of more traditional quantum circuits.

We develop a framework for analyzing layered quantum algorithms such as quantum alternating operator ans\"atze. Our framework relates quantum cost gradient operators, derived from the cost and mixing Hamiltonians, to classical cost difference functions that reflect cost function neighborhood structure. By considering QAOA circuits from the Heisenberg picture, we derive exact general expressions for expectation values as series expansions in the algorithm parameters, cost gradient operators, and cost difference functions. This enables novel interpretability and insight into QAOA behavior in various parameter regimes. For single-level QAOA1 we show the leading-order changes in the output probabilities and cost expectation value explicitly in terms of classical cost differences, for arbitrary cost functions. This demonstrates that, for sufficiently small positive parameters, probability flows from lower to higher cost states on average. By selecting signs of the parameters, we can control the direction of flow. We use these results to derive a classical random algorithm emulating QAOA1 in the small-parameter regime, i.e., that produces bitstring samples with the same probabilities as QAOA1 up to small error. For deeper QAOAp circuits we apply our framework to derive analogous and additional results in several settings. In particular we show QAOA always beats random guessing. We describe how our framework incorporates cost Hamiltonian locality for specific problem classes, including causal cone approaches, and applies to QAOA performance analysis with arbitrary parameters. We illuminate our results with a number of examples including applications to QUBO problems, MaxCut, and variants of MaxSat. We illustrate the application to QAOA circuits using mixing unitaries beyond the transverse-field mixer through two examples of constrained optimization, Max Independent Set and Graph Coloring.

Quantum optimization, both for classical and quantum functions, is one of the most well-studied applications of quantum computing, but recent trends have relied on hybrid methods that push much of the fine-tuning off onto costly classical algorithms. Feedback-based quantum algorithms, such as FALQON, avoid these fine-tuning problems but at the cost of additional circuit depth and a lack of convergence guarantees. In this work, we take the local greedy information collected by Lyapunov feedback control and develop an analytic framework to use it to perturbatively update previous control layers, similar to the global optimal control achievable using Pontryagin optimal control. This perturbative methodology, which we call Feedback Optimally Controlled Quantum States (FOCQS), can be used to improve the results of feedback-based algorithms, like FALQON. Furthermore, this perturbative method can be used to push smooth annealing-like control protocol closer to the control optimum, even providing and iterative approach, albeit with diminishing returns. In numerical testing, we show improvements in convergence and required depth due to these methods over existing quantum feedback control methods.

We consider the power of local algorithms for approximately solving Max \\(k\\)XOR, a generalization of two constraint satisfaction problems previously studied with classical and quantum algorithms (MaxCut and Max E3LIN2). In Max \\(k\\)XOR each constraint is the XOR of exactly \\(k\\) variables and a parity bit. On instances with either random signs (parities) or no overlapping clauses and \\(D+1\\) clauses per variable, we calculate the expected satisfying fraction of the depth-1 QAOA from Farhi et al [arXiv:1411.4028] and compare with a generalization of the local threshold algorithm from Hirvonen et al [arXiv:1402.2543]. Notably, the quantum algorithm outperforms the threshold algorithm for \\(k > 4\\). On the other hand, we highlight potential difficulties for the QAOA to achieve computational quantum advantage on this problem. We first compute a tight upper bound on the maximum satisfying fraction of nearly all large random regular Max \\(k\\)XOR instances by numerically calculating the ground state energy density \\(P(k)\\) of a mean-field \\(k\\)-spin glass [arXiv:1606.02365]. The upper bound grows with \\(k\\) much faster than the performance of both one-local algorithms. We also identify a new obstruction result for low-depth quantum circuits (including the QAOA) when \\(k=3\\), generalizing a result of Bravyi et al [arXiv:1910.08980] when \\(k=2\\). We conjecture that a similar obstruction exists for all \\(k\\).