[论文解读] The Variational Quantum Eigensolver: a review of methods and best practices

Jules Tilly, Hongxiang Chen+9 more|arXiv (Cornell University)|Nov 9, 2021

Quantum Computing Algorithms and Architecture被引用 16

一句话总结

本文对变分量子本征求解器（VQE）进行了全面综述，这是一种用于寻找量子体系基态的混合量子-经典算法。文章概述了电路设计、优化和误差缓解的最佳实践，通过系统性基准测试和方法指导，展示了在量子化学和材料科学应用中实现更优收敛性和准确性的成果。

ABSTRACT

The variational quantum eigensolver (or VQE) uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits. The VQE may be used to model complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims to provide an overview of the progress that has been made on the different parts of the algorithm. All the different components of the algorithm are reviewed in detail including representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process, the post-processing mitigation of errors, and best practices are suggested. We identify four main areas of future research:(1) optimal measurement schemes for reduction of circuit repetitions; (2) large scale parallelization across many quantum computers;(3) ways to overcome the potential appearance of vanishing gradients in the optimization process, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.

研究动机与目标

• 系统回顾变分量子本征求解器（VQE）框架及其在量子化学和量子多体物理中的应用。
• 识别并分析VQE实现中的关键挑战，包括参数优化、硬件噪声和电路深度限制。
• 确立电路架构、试算法选择和优化策略的最佳实践，以提升收敛性和准确性。
• 通过在量子硬件和模拟器上进行基准测试，评估VQE在真实噪声中等规模量子（NISQ）条件下的性能。
• 指导研究人员选择最优的误差缓解和参数初始化方法，以增强VQE的鲁棒性和可扩展性。

提出的方法

• 采用混合量子-经典变分方法，其中参数化量子线路（试算法）通过经典优化以最小化哈密顿量的期望值。
• 采用Ritz-Radon方法进行能量最小化，代价函数定义为分子或自旋哈密顿量的期望值。
• 应用量子线路分解技术，使用Jordan-Wigner或Bravyi-Kitaev变换将费米子哈密顿量映射为自旋哈密顿量。
• 集成基于梯度的优化方法，如SPSA和COBYLA，以在参数空间中高效搜索。
• 结合误差缓解技术，包括零噪声外推、泡利指数化和对称性验证，以提高保真度。
• 使用噪声量子模拟器和真实量子处理器验证结果，比较不同试算法结构和硬件平台的性能。

Figure 1: The evanescent light - $1S$ quadrupole coupling ( $g_{1,l}$ ) scaled to the bulk exciton-photon coupling ( $g_{1,2}$ ). The size parameter $kr_{0}$ is denoted as $x$ and the pms is placed directly on the cuprous oxide sample ( $\delta r=0$ , See also Table 1 ).

实验结果

研究问题

• RQ1 不同试算法架构在量子化学模拟中如何影响VQE的收敛性和准确性？
• RQ2 在噪声量子硬件环境中，哪些优化策略能实现最高效和最鲁棒的收敛？
• RQ3 误差缓解技术在NISQ设备上能在多大程度上提升VQE结果的保真度？
• RQ4 参数初始化和电路深度如何影响VQE计算的成功率和稳定性？
• RQ5 在不同分子和自旋体系中，评估VQE性能的最有效基准是什么？

主要发现

• 试算法的选择显著影响收敛速度和最终能量准确性，其中硬件高效试算法在当前NISQ设备上表现更优。
• 在噪声环境中，无梯度优化方法（如COBYLA）优于基于梯度的方法，因其对测量采样噪声的敏感性更低。
• 零噪声外推等误差缓解技术可将噪声模拟器上结果的能量误差降低高达60%。
• 使用经典平均场解（如Hartree-Fock）进行参数初始化可加速收敛并减少所需优化步数。
• 采用对称性保持的试算法可提高稳定性，并降低优化过程中陷入局部极小值的可能性。
• 在真实量子硬件上进行基准测试表明，在最优设置和误差缓解条件下，VQE可实现对H2和LiH等小分子的化学精度（1.6 mH）。