[论文解读] Quantum Algorithms for Scientific Computing and Approximate Optimization

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

Quantum Computing Algorithms and Architecture参考文献 211被引用 31

一句话总结

本文提出了用于科学计算和组合优化的新型量子算法，引入了用于平方根、对数等基本函数的模块化量子线路，一种具有多项式缩放特性的微扰量子算法以近似哈密顿量的低能本征值，以及一种适用于具有可行性约束的优化问题的广义量子交替算符试探法，显著提升了近期量子设备的资源效率。

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.

研究动机与目标

• 为诸如平方根、对数和分数幂等基本数值函数，开发模块化、可逆且具有误差控制的量子算法。
• 设计一种高效的量子算法，用于近似量子多体系统中哈密顿量的低阶本征值，克服经典方法的指数级缩放问题。
• 通过利用群特异性结构的分治策略，改进哈密顿量模拟，降低模拟成本。
• 通过广义框架将量子近似优化算法（QAOA）扩展至处理具有可行性约束的组合优化问题。
• 为关键的NP难问题（如最大独立集、图着色问题和旅行商问题）提供具体的量子线路构造与资源估算。

提出的方法

• 使用具有有界最坏情况误差和可逆门分解的迭代逼近技术，设计用于基本函数的量子线路。
• 应用微扰方法估计哈密顿量的低阶本征值，利用受控误差和成功概率的量子相位估计算法。
• 通过将哈密顿量划分为子群，对每个子群使用定制化技术进行模拟，并通过量子线路组合将结果合并，实现哈密顿量模拟的分治策略。
• 通过引入非阿贝尔和非移动的酉算符，将QAOA广义化为量子交替算符试探法，以更好地编码问题约束。
• 开发一种系统化程序，将组合优化问题映射为量子线路，包括最大独立集、图着色和TSP的显式构造。
• 在对量子化学应用的温和假设下，基于系统规模、期望精度和问题特异性参数，推导所有算法的成本界限。

实验结果

研究问题

• RQ1 能否为科学计算任务（如计算平方根和对数）设计出具有模块化、可逆性和数值稳定性的量子算法？
• RQ2 能否设计一种量子算法，以系统规模和逆精度的多项式缩放方式近似哈密顿量的低阶本征值，从而超越经典方法的指数级缩放？
• RQ3 通过利用子哈密顿量的结构特性，分治方法能否在哈密顿量模拟中获得更紧致的成本界限？
• RQ4 QAOA框架能否被广义化以处理具有可行性约束的组合优化问题？在电路深度增加时，性能权衡如何？
• RQ5 使用新试探法将典型NP难问题映射到量子线路时，其资源成本和线路实现是什么？

主要发现

• 本文构建了用于计算平方根、对数和任意分数幂的量子线路，具有有界最坏情况误差和高效的门计数。
• 所提出的微扰量子算法以系统规模和逆精度的多项式成本缩放，近似哈密顿量的常数个低阶本征值，且成功概率较高。
• 在温和假设下，分治哈密顿量模拟方法显著改进了量子化学中电子结构问题的成本估计。
• 广义量子交替算符试探法使受约束优化问题的低资源实现成为可能，且为最大独立集、图着色和旅行商问题提供了显式线路构造。
• 针对最大割问题的最低深度QAOA实现，推导出解析性能界限，证明了其在一般图类和特殊图类上的性能。
• 研究揭示了将解析性能界限扩展至浅层电路深度之外的固有挑战，凸显了当前解析技术在深层QAOA电路中的局限性。