Estimating Trotter Approximation Errors to Optimize Hamiltonian Partitioning for Lower Eigenvalue Errors

TL;DR

This paper evaluates different methods for estimating Trotter approximation errors in quantum simulations, finding perturbation theory-based estimates correlate well with actual errors, unlike norm-based bounds, aiding in resource optimization.

Contribution

It introduces a perturbation theory-based error estimator for Trotter approximation that correlates strongly with true errors, improving Hamiltonian partitioning strategies.

Findings

Norm-based estimators show low correlation (<0.4) with true errors.

Perturbation theory estimates exhibit high correlation with actual Trotter errors.

The results support using perturbative estimates for practical quantum resource management.

Abstract

Trotter approximation in conjunction with Quantum Phase Estimation can be used to extract eigen-energies of a many-body Hamiltonian on a quantum computer. There were several ways proposed to assess the quality of this approximation based on estimating the norm of the difference between the exact and approximate evolution operators. Here, we explore how different error estimators correlate with the true error in the ground state energy due to Trotter approximation. For a set of small molecules we calculate these exact error in ground-state electronic energies due to the second-order Trotter approximation. Comparison of these errors with previously used upper bounds show correlation less than 0.4 across various Hamiltonian partitionings. On the other and, building the Trotter approximation error estimation based on perturbation theory up to a second order in the time-step for eigenvalues provides estimates with very good correlations with the exact Trotter approximation errors. These findings highlight the non-faithful character of norm-based estimations for prediction of a Trotter-based eigenvalue estimation performance and the need of alternative estimators. The developed perturbative estimates can be used for practical time-step and Hamiltonian partitioning selection protocols, which are needed for an accurate assessment of quantum resources.