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

• Main Authors: Jones, Benjamin D. M, Mineh, Lana, Montanaro, Ashley
• Format: Journal Article
• Language: English
• Published: 20-11-2024
• Subjects: Computer Science - Learning; Computer Science - Neural and Evolutionary Computing; Physics - Quantum Physics
• DOI: 10.48550/arxiv.2411.13742

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.