Trotterization, Operator Scrambling, and Entanglement

TL;DR

This paper establishes a fundamental link between operator scrambling, entanglement, and Trotter errors in quantum simulations, providing refined bounds and insights to improve simulation accuracy and efficiency.

Contribution

It reveals the connection between operator scrambling and Trotter error bounds, and analyzes how entanglement influences simulation robustness and error scaling.

Findings

Trotter error is bounded by operator scrambling degree.

Entanglement can suppress simulation errors even at low levels.

Error scaling depends on normalized Frobenius norms of observables and errors.

Abstract

Operator scrambling, which governs the spread of quantum information in many-body systems, is a central concept in both condensed matter and high-energy physics. Accurately capturing the emergent properties of these systems remains a formidable challenge for classical computation, while quantum simulators have emerged as a powerful tool to address this complexity. In this work, we reveal a fundamental connection between operator scrambling and the reliability of quantum simulations. We show that the Trotter error in simulating operator dynamics is bounded by the degree of operator scrambling, providing the most refined analysis of Trotter errors in operator dynamics so far. Furthermore, we investigate the entanglement properties of the evolved states, revealing that sufficient entanglement can lead to error scaling governed by the normalized Frobenius norms of both the observables of interest and the error operator, thereby enhancing simulation robustness and efficiency compared to previous works. We also show that even in regimes where the system's entanglement remains low, operator-induced entanglement can still emerge and suppress simulation errors. Our results unveil a comprehensive relationship between Trotterization, operator scrambling, and entanglement, offering new perspectives for optimizing quantum simulations.