Classical Benchmarks for Variational Quantum Eigensolver Simulations of the Hubbard Model

TL;DR

This study benchmarks the accuracy of variational quantum algorithms for simulating the Hubbard model on classical computers, revealing limitations in energy accuracy for larger systems and the effects of correlations and inhomogeneity.

Contribution

It provides a comprehensive classical benchmarking of variational quantum eigensolvers applied to the Hubbard model, analyzing factors affecting accuracy and system size dependence.

Findings

Error plateaus for larger lattices even with optimal ansatzes

Stronger correlations increase energy errors

Inhomogeneity and off-site interactions have minimal impact

Abstract

Simulating the Hubbard model is of great interest to a wide range of applications within condensed matter physics, however its solution on classical computers remains challenging in dimensions larger than one. The relative simplicity of this model, embodied by the sparseness of the Hamiltonian matrix, allows for its efficient implementation on quantum computers, and for its approximate solution using variational algorithms such as the variational quantum eigensolver. While these algorithms have been shown to reproduce the qualitative features of the Hubbard model, their quantitative accuracy in terms of producing true ground state energies and other properties, and the dependence of this accuracy on the system size and interaction strength, the choice of variational ansatz, and the degree of spatial inhomogeneity in the model, remains unknown. Here we present a rigorous classical benchmarking study, demonstrating the potential impact of these factors on the accuracy of the variational solution of the Hubbard model on quantum hardware, for systems with up to $32$ qubits. We find that even when using the most accurate wavefunction ans\"{a}tze for the Hubbard model, the error in its ground state energy and wavefunction plateaus for larger lattices, while stronger electronic correlations magnify this issue. Concurrently, spatially inhomogeneous parameters and the presence of off-site Coulomb interactions only have a small effect on the accuracy of the computed ground state energies. Our study highlights the capabilities and limitations of current approaches for solving the Hubbard model on quantum hardware, and we discuss potential future avenues of research.