Trotter error with commutator scaling for the Fermi-Hubbard model

We derive higher-order error bounds with small prefactors for a general Trotter product formula, generalizing a result given by Childs et al. [Phys. Rev. X 11, 011020 (2021)]. We then apply these bounds to the real-time quantum time evolution operator governed by the Fermi-Hubbard Hamiltonian on one-dimensional and two- dimensional square and triangular lattices. The main technical contribution of our work is a symbolic evaluation of nested commutators between hopping and interaction terms for a given lattice geometry. The calculations result in explicit expressions for the error bounds in terms of the time step and Hamiltonian coefficients. Comparison with the actual Trotter error (evaluated on a small system) indicates that the bounds still overestimate the error.

Automated summary

This paper derives higher-order error bounds with small prefactors for a general Trotter product formula, generalizing a result given by Childs et al. [Phys. Rev. X 11, 011020 (2021)]. The bounds are then applied to the real-time quantum time evolution operator governed by the Fermi-Hubbard Hamiltonian on one-dimensional and two-dimensional square and triangular lattices. The main technical contribution of the work is a symbolic evaluation of nested commutators between hopping and interaction terms for a given lattice geometry. However, comparison with the actual Trotter error (evaluated on a small system) shows that the bounds still overestimate the error.