[Paper Review] Quantum Algorithms for Scientific Computing and Approximate Optimization

Stuart Hadfield|arXiv (Cornell University)|Jan 1, 2018

Quantum Computing Algorithms and ArchitectureComputer Science211 references31 citations

TL;DR

This paper presents novel quantum algorithms for scientific computing and combinatorial optimization, introducing modular quantum circuits for elementary functions like square roots and logarithms, a perturbative quantum algorithm for approximating low-energy eigenvalues of Hamiltonians with polynomial scaling, and a generalized Quantum Alternating Operator Ansatz for optimization problems with feasibility constraints, achieving improved resource efficiency for near-term quantum devices.

ABSTRACT

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.

Motivation & Objective

• To develop modular, reversible, and error-controlled quantum algorithms for fundamental numerical functions such as square roots, logarithms, and fractional powers.
• To design an efficient quantum algorithm for approximating low-order eigenvalues of Hamiltonians in quantum many-body systems, overcoming exponential classical scaling.
• To improve Hamiltonian simulation through a divide-and-conquer approach that exploits group-specific structures to reduce simulation costs.
• To extend the Quantum Approximate Optimization Algorithm (QAOA) to handle combinatorial problems with feasibility constraints via a generalized framework.
• To provide concrete quantum circuit constructions and resource estimates for key NP-hard problems like Maximum Independent Set, Graph Coloring, and the Traveling Salesman Problem.

Proposed method

• Design quantum circuits for elementary functions using iterative approximation techniques with bounded worst-case error and reversible gate decomposition.
• Apply a perturbative approach to estimate low-order eigenvalues of a Hamiltonian, leveraging quantum phase estimation with controlled error and success probability.
• Implement a divide-and-conquer strategy for Hamiltonian simulation by partitioning the Hamiltonian into subgroups, simulating each group with tailored techniques, and combining results via quantum circuit composition.
• Generalize QAOA by introducing the Quantum Alternating Operator Ansatz, which uses non-Abelian and non-traveling unitary operators to better encode problem constraints.
• Develop a systematic procedure to map combinatorial optimization problems to quantum circuits, including explicit constructions for Maximum Independent Set, Graph Coloring, and TSP.
• Derive cost bounds for all algorithms based on system size, desired accuracy, and problem-specific parameters, under mild assumptions for quantum chemistry applications.

Experimental results

Research questions

• RQ1 Can quantum algorithms be designed with modular, reversible, and numerically stable components for scientific computing tasks such as computing square roots and logarithms?
• RQ2 Can a quantum algorithm approximate low-order eigenvalues of a Hamiltonian with polynomial scaling in system size and inverse accuracy, outperforming classical exponential scaling?
• RQ3 Can a divide-and-conquer approach to Hamiltonian simulation yield tighter cost bounds by exploiting structural properties of sub-Hamiltonians?
• RQ4 Can the QAOA framework be generalized to handle combinatorial optimization problems with feasibility constraints, and what are the performance trade-offs at increasing circuit depth?
• RQ5 What are the resource costs and circuit implementations for mapping prototypical NP-hard problems to quantum circuits using the new ansatz?

Key findings

• The paper constructs quantum circuits for computing square roots, logarithms, and arbitrary fractional powers with bounded worst-case error and efficient gate counts.
• The proposed perturbative quantum algorithm approximates a constant number of low-order eigenvalues of a Hamiltonian with cost scaling polynomially in system size and inverse accuracy, achieving high success probability.
• The divide-and-conquer Hamiltonian simulation method yields significantly improved cost estimates for the electronic structure problem in quantum chemistry under mild assumptions.
• The generalized Quantum Alternating Operator Ansatz enables low-resource implementations for constrained optimization problems, with explicit circuit constructions for Maximum Independent Set, Graph Coloring, and the Traveling Salesman Problem.
• Analytic performance bounds are derived for the lowest-depth QAOA realization on the Maximum Cut problem, demonstrating provable performance on both general and special graph classes.
• The study reveals inherent challenges in extending analytic performance bounds beyond shallow circuit depths, highlighting limitations in current analytical techniques for deeper QAOA circuits.