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：我们针对采用哈密顿量变分拟设求解费米-哈伯德系统的变分量子本征求解器，在372个算例上对30种优化器进行了数值基准测试。根据达到的最终能量值及进入特定容差范围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动量梯度下降、ADAM（采用有限差分法）、SPSA、CMAES和BayesMGD等梯度下降变体。通过梯度分析发现有限差分法的步长设置具有显著影响。我们还研究了采用同步扰动（受SPSA启发）作为梯度子程序的方法：有限差分法能获得更精确的基态估计但需要更多函数调用，而同步扰动法收敛更快但在后期阶段精度可能降低。最后，我们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，虽然该算法能以更少迭代次数达到更低能量，但综合考虑总函数调用次数后该优势通常消失。本研究通过对4个算例进行精细超参数扫描，提供了多维度分析图表、详细优化器注释，并探讨了未来研究方向。