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Prethermalization, Shadowing Breakdown, and the Absence of Trotterization Transition in Quantum Circuits

Marko Žnidarič

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Phys. Rev. X 16, 021017 – Published 22 April, 2026

DOI: https://doi.org/10.1103/69d7-83xh

Abstract

One of the premier utilities of present-day noisy quantum computers is simulation of many-body quantum systems. We study how long in time is such a discrete-time simulation representative of a continuous time Hamiltonian evolution; namely, a finite time step introduces so-called Trotterization errors. We demonstrate that the truncated operator propagator (Ruelle-Pollicott resonances) is a powerful tool to that end, as well as to study prethermalization and discrete-time crystals, including finding those phenomena at large gate duration. We show that the effective energy is more stable than suggested by Trotter errors—a manifestation of prethermalization—while all other observables are not. Even the most stable observable though deteriorates in the thermodynamic limit. Different than in classical systems with the strongest chaos, where the faithfulness time (the shadowing time) can be infinite, in quantum many-body chaotic systems it is finite. A corollary of our results is also that, opposite of previous claims, there is no Trotterization transition in nonintegrable many-body quantum systems. We demonstrate our results on a one-dimensional (1D) kicked Ising model, as well as on a 1D kicked XX model and a 2D kicked Ising model. The truncated propagator is also used to calculate the energy diffusion constant in the tilted-field Ising model with high accuracy.

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Simulating many-body quantum systems on noisy computers introduces Trotterization errors that can limit the faithfulness of the simulation. We study how long these discrete-time simulations remain representative of continuous Hamiltonian evolution by using a truncated operator propagator. We find that while the effective energy is remarkably stable due to prethermalization, the shadowing time for all observables in chaotic quantum systems is ultimately finite. Our results further demonstrate that previously reported Trotterization transitions in nonintegrable systems are actually finite-size artifacts that disappear in the thermodynamic limit. This methodology provides a high-precision tool for calculating transport properties and assessing the long-term stability of quantum simulations, thereby establishing rigorous limits for quantum supremacy demonstrations by calculating the timescales upon which digital simulators fundamentally diverge from the systems they mimic.

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Phys. Rev. X 16, 021017– Published 22 April, 2026
  • Received 27 May 2025
  • Revised 12 November 2025
  • Accepted 17 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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