Lindbladian Simulation with Commutator Bounds

TL;DR

This paper develops new commutator-based error bounds for Lindbladian simulation using Trotter decomposition, improving system-size scaling and precision in open quantum system simulations.

Contribution

It derives novel Trotter error bounds for Lindbladian dynamics, including bounds for observable estimation and a truncation method for the BCH expansion.

Findings

Achieves $O(\sqrt{N})$ Trotter step scaling for local systems

Uses Richardson extrapolation for polylogarithmic precision

Numerical results confirm the theoretical scaling improvements

Abstract

Trotter decomposition provides a simple approach to simulating open quantum systems by decomposing the Lindbladian into a sum of individual terms. While it is established that Trotter errors in Hamiltonian simulation depend on nested commutators of the summands, such a relationship remains poorly understood for Lindbladian dynamics. In this Letter, we derive commutator-based Trotter error bounds for Lindbladian simulation, yielding an $O(\sqrt{N})$ scaling in the number of Trotter steps for locally interacting systems on $N$ sites. When estimating observable averages, we apply Richardson extrapolation to achieve polylogarithmic precision while maintaining the commutator scaling. To bound the extrapolation remainder, we develop a general truncation bound for the Baker-Campbell-Hausdorff expansion that bypasses common convergence issues in physically relevant systems. For local Lindbladians, our results demonstrate that the Trotter-based methods outperform prior simulation techniques in system-size scaling while requiring only $O(1)$ ancillas. Numerical simulations further validate the predicted system-size and precision scaling.