We numerically benchmark 30 optimisers on 372 instances of the variational quantum eigensolver for solving the Fermi-Hubbard system with the Hamiltonian variational ansatz. We rank the optimisers with respect to metrics such as final energy achieved and function calls needed to get within a certain tolerance level, and find that the best performing optimisers are variants of gradient descent such as Momentum and ADAM (using finite difference), SPSA, CMAES, and BayesMGD. We also perform gradient analysis and observe that the step size for finite difference has a very significant impact. We also consider using simultaneous perturbation (inspired by SPSA) as a gradient subroutine: here finite difference can lead to a more precise estimate of the ground state but uses more calls, whereas simultaneous perturbation can converge quicker but may be less precise in the later stages. Finally, we also study the quantum natural gradient algorithm: we implement this method for 1-dimensional Fermi-Hubbard systems, and find that whilst it can reach a lower energy with fewer iterations, this improvement is typically lost when taking total function calls into account. Our method involves performing careful hyperparameter sweeping on 4 instances. We present a variety of analysis and figures, detailed optimiser notes, and discuss future directions.

翻译：我们通过数值方法对30种优化器在372个变分量子本征求解器实例上进行了基准测试，这些实例均采用哈密顿变分拟设求解费米-哈伯德系统。我们根据达到的最终能量值及在特定容差范围内所需的函数调用次数等指标对优化器进行排序，发现表现最佳的优化器包括梯度下降的变体（如Momentum和ADAM（使用有限差分法））、SPSA、CMAES以及BayesMGD。我们还进行了梯度分析，观察到有限差分的步长设置具有极其显著的影响。同时，我们探索了将同步扰动（受SPSA启发）作为梯度子程序的方法：有限差分法能够更精确地估计基态能量但需要更多函数调用，而同步扰动法收敛速度更快但在后期阶段可能精度较低。最后，我们还研究了量子自然梯度算法：在一维费米-哈伯德系统中实施该方法时发现，虽然该算法能够以更少的迭代次数达到更低的能量值，但当计入总函数调用次数时，这种优势通常不复存在。我们的研究方法包括对4个实例进行细致的超参数扫描。本文提供了多维度分析与图表展示、详细的优化器说明，并对未来研究方向进行了探讨。