Trotter Scars: Trotter Error Suppression in Quantum Simulation

TL;DR

This paper introduces Trotter scars, special initial states with suppressed Trotter errors in quantum simulation, and provides a variational method to find such states, enhancing simulation accuracy.

Contribution

It analytically explains the spectral origin of Trotter-error suppression and offers a practical framework to identify error-resilient states in quantum systems.

Findings

Trotter scars exhibit strongly suppressed error growth.

Spectrally commensurate states show persistent Loschmidt revivals.

The variational framework successfully finds optimized error-minimizing states.

Abstract

Recent studies have shown that Trotter errors are highly initial-state dependent and that standard upper bounds often substantially overestimate them. However, the mechanism underlying anomalously small Trotter errors and a systematic route to identifying error-resilient states remain unclear. Using interaction-picture perturbation theory, we derive an analytical expression for the leading-order Trotter error in the eigenbasis of the Hamiltonian. Our analysis shows that initial states supported on spectrally commensurate energy ladders exhibit strongly suppressed error growth together with persistent Loschmidt revivals. We refer to such states as Trotter scars. To identify such states in practice, we further introduce a general variational framework for finding error-minimizing initial states for a given Hamiltonian. Applying this framework to several spin models, we find optimized states whose spectral support and dynamical behavior agree with the perturbative prediction. Our results reveal the spectral origin of Trotter-error resilience and provide a practical strategy for discovering error-resilient states in digital quantum simulation.