# Nearly optimal lattice simulation by product formulas

## TL;DR

This paper proves that product formulas can simulate lattice Hamiltonians on quantum computers with nearly optimal gate complexity, providing rigorous error bounds and extending previous results.

## Contribution

It offers a rigorous proof of the nearly optimal gate complexity for lattice Hamiltonian simulation using product formulas, including error analysis and generalizations.

## Key findings

- Gate complexity is (nt)^{1+o(1)} for lattice Hamiltonian simulation.
- Provides error bounds for canonical product formulas including Lie-Trotter-Suzuki.
- Extends analysis to time-dependent Hamiltonians and higher dimensions.

## Abstract

We consider simulating an $n$-qubit Hamiltonian with nearest-neighbor interactions evolving for time $t$ on a quantum computer. We show that this simulation has gate complexity $(nt)^{1+o(1)}$ using product formulas, a straightforward approach that has been demonstrated by several experimental groups. While it is reasonable to expect this complexity---in particular, this was claimed without rigorous justification by Jordan, Lee, and Preskill---we are not aware of a straightforward proof. Our approach is based on an analysis of the local error structure of product formulas, as introduced by Descombes and Thalhammer and further simplified here. We prove error bounds for canonical product formulas, which include well-known constructions such as the Lie-Trotter-Suzuki formulas. We also develop a local error representation for time-dependent Hamiltonian simulation, and we discuss generalizations to periodic boundary conditions, constant-range interactions, and higher dimensions. Combined with a previous lower bound, our result implies that product formulas can simulate lattice Hamiltonians with nearly optimal gate complexity.