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

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.