Benchmarking a wide range of optimisers for solving the Fermi-Hubbard model using the variational quantum eigensolver

TL;DR

The paper systematically benchmarks 30 optimisers for solving the Fermi-Hubbard problem with a Hamiltonian variational (HV) ansatz within VQE across 372 instances, evaluating both final ground-state energy and total cost-call efficiency. It analyzes gradient strategies (finite differences vs simultaneous perturbation) and includes a dedicated look at the quantum natural gradient in 1D FH systems, revealing that improvements in iteration count often do not translate into fewer total calls. Key findings show Momentum or Adam with finite-difference gradients typically yield the best final energies, while SPSA, CMAES, and BayesMGD excel when the aim is fewer function evaluations; hyperparameter tuning significantly impacts performance. The study provides practical guidance for optimiser selection on HV-VQE FH problems and offers a public dataset for further benchmarking, highlighting potential multi-stage and hybrid strategies for future work.

Abstract

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.

Figures (18)

• Figure 1: Boxplots of optimisers tested over all 372 VQE for Fermi-Hubbard instances, ordered in terms of performance, i.e. average normalised (divided by the size of the grid $m \times n$) final energy difference with the ground energy. The suffices 'fd' and 'sp' refer to the finite difference and simultaneous perturbation gradient subroutines respectively. See \ref{['fig:boxplot all instances']} and \ref{['sec:results']} for more discussion.
• Figure 2: Plot of finite difference step size with absolute error (norm of difference) with the exact gradient. We see a trade-off between the error introduced by finite difference, and the error appearing from the statistical noise.
• Figure 3: Gradient sweeping: boxplots of best step size found for 100 random points on four instances that minimises the error with the exact gradient. The x axis denotes the best step size found for each of the 100 random points. From this we use a step size of 0.4 in numerical simulations, as the average of the four means to 1 decimal place.
• Figure 4: We ran BFGS, L-BFGS-B, Nelder-Mead, Powell and SLSQP (from scipyscipy) on exact cost functions for the instances considered, to determine the expressivity of the ansätze in the noiseless case. We plot the closest final energy to the ground state. Here "occ" refers to the occupation number, i.e. the sum of spin up and spin down particles (which we set to be the same) -- these values correspond to quarter and half filling.
• Figure 5: Plots of individual runs, displaying the exact energy throughout a run (although the optimisers only have access to a statistical cost function).