A Theory of Trotter Error

TL;DR

This paper develops a comprehensive theory of Trotter error that provides tighter bounds and improved algorithms for quantum simulation, applicable to a wide range of systems and interactions, surpassing previous limitations.

Contribution

It introduces a general, commutativity-exploiting framework for analyzing Trotter error, leading to better bounds and algorithms for quantum simulation of diverse Hamiltonians.

Findings

Tighter error bounds for real- and imaginary-time evolutions.

Nearly matching or outperforming previous algorithms for various systems.

Local observables can be simulated with complexity independent of system size in power-law systems.

Abstract

The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, $k$-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of $5$, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.