Trotter Scars: Trotter Error Suppression in Quantum Simulation

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, we further introduce a model-agnostic variational framework. Its loss function can be built from Trotterized dynamics alone, which allows the search to reach system sizes beyond exact diagonalization. The optimized states at small sizes moreover follow regular patterns that extend to larger sizes. We demonstrate our theory in three spin models, where the optimized states exhibit the predicted persistent Loschmidt revivals and strongly suppressed error growth. We further conducted experiments on a $17$-qubit superconducting quantum processor and successfully realized the Trotter-scar states and demonstrated the Trotter error suppression in quantum simulations.