Lindbladian Simulation with Commutator Bounds

2026-03-30

AI Summary

This work addresses the absence of rigorous commutator-based error bounds for Trotter decompositions of Lindbladian dynamics in open quantum systems. It establishes, for the first time, a quantitative relationship between Trotter errors and nested commutators, yielding a truncation error bound for the Baker–Campbell–Hausdorff expansion tailored to local Lindbladian systems. By integrating Richardson extrapolation, the method enables efficient simulation of observable expectation values using only a constant number of ancillary qubits. Theoretical analysis demonstrates that the algorithm achieves a Trotter step scaling of $O(\sqrt{N})$ with polylogarithmic dependence on precision, surpassing existing approaches. Numerical experiments corroborate the predicted scaling behavior.

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.

Problem

• Lindbladian simulation
• Trotter error
• commutator bounds
• open quantum systems
• quantum simulation

Innovation

• Lindbladian simulation
• Trotter error bounds
• commutator scaling
• Richardson extrapolation
• open quantum systems