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arXiv:2505.15521 · Physical Review X

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

Marko Žnidarič

Published Mar 17, 2026• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/69d7-83xh

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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.
We study universal aspects of thermalization induced by Trotterization, a procedure routinely used in gate-based quantum computation. We use the reduced-Bardeen-Cooper-Schrieffer model-quantum integrable with a classically integrable mean-field limit-where the effects of Trotter chaos are expected to be particularly stark. The resulting Trotterized chaotic dynamics is characterized by its Lyapunov spectrum and rescaled Kolmogorov-Sinai entropy. The chaos quantifiers depend on the Trotterization time step τ. We observe a Trotter transition at a finite step value τ_{c}≈sqrt[N]. While the dynamics is weakly chaotic for time steps τ≪τ_{c}, the regime of large Trotterization steps is characterized by short temporal correlations. We derive two different scaling laws for the two different regimes by numerically fitting the maximum Lyapunov exponent data. The scaling law of the large τ limit agrees well with the one derived from the kicked top map. Beyond its relevance to current quantum computers, our work opens other directions-such as probing observables like the Loschmidt echo, which lie beyond standard mean-field description--across the Trotter transition we uncover.
Quantum simulation is one of the most promising scientific applications of quantum computers.Due to decoherence and noise in current devices, it is however challenging to perform digital quantum simulation in a regime that is intractable with classical computers.In this work, we propose an experimental protocol for probing dynamics and equilibrium properties on near-term digital quantum computers.As a key ingredient of our work, we show that it is possible to study thermalization even with a relatively coarse Trotter decomposition of the Hamiltonian evolution of interest.Even though the step size is too large to permit a rigorous bound on the Trotter error, we observe that the system prethermalizes in accordance with previous results for Floquet systems.The dynamics closely resemble the thermalization of the model underlying the Trotterization up to long times.We make our approach resilient to noise by developing an error mitigation scheme based on measurement and rescaling of survival probabilities, which is applicable to time-evolution problems in general.We demonstrate the effectiveness of the entire protocol by applying it to the two-dimensional XY model and we numerically verify its performance with realistic noise parameters for superconducting quantum devices.Our proposal thus provides a route to achieving quantum advantage for relevant problems in condensed-matter physics.
A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.
We describe a chaos scenario valid for chaotic many-body quantum systems, the simulation of which is one of the prime candidates for demonstrating quantum advantage, and clarify its importance for classical truncation-based simulations. By studying the spectral properties of the truncated propagator, we, for the first time, identify fractal structures responsible for many-body quantum chaos, analogous to the fractality in classical chaos emerging from the time evolution of initially smooth long-wavelength densities into ever finer short-wavelength densities. Namely, we show that the leading slowest-decaying Ruelle-Pollicott eigenoperators have a non-trivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes an asymptotic equality between the temporal decay rate of local correlations and this many-body fractal dimension. With this insight, the scenario for many-body quantum chaos becomes clear: over time, local operators evolve toward increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators -- a kind of many-body Kolmogorov cascade in the space of operators -- which enables accurate truncation-based simulations. Our predictions are demonstrated numerically in various one- and two-dimensional quantum circuits, with and without conservation laws, as well as by exact results in dual-unitary circuits.
A fundamental challenge in digital quantum simulation (DQS) is the control of an inherent error, which appears when discretizing the time evolution of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables, leading to an error independent of system size and simulation time. DQS is thus intrinsically much more robust than suggested by known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size, which separates a localized region with controllable Trotter errors from a quantum chaotic regime. Our findings show that DQS with comparatively large Trotter steps can retain controlled errors for local observables. It is thus possible to reduce the number of gate operations required to represent the desired time evolution faithfully.
Digital quantum simulation (DQS) is one of the most promising applications of quantum computing. Trotterised DQS models quantum dynamics by decomposing the evolution into a sequence of gate-based Trotter steps. Using larger Trotter steps reduces the required gate count, but it is now understood that the Trotterisation step size cannot be increased beyond a certain threshold value, beyond which there is a sudden breakdown in simulation accuracy, due to the onset of quantum chaos. To date, this behaviour has only been studied for closed systems not subjected to the environmental noise and decoherence to which real-world quantum computers are highly sensitive. In this work, we study a range of experimentally relevant open-system quantum simulator models and show that Trotterisation thresholds persist even when open-system effects are included, even in the long-time, steady-state limit, once all coherence has been lost. We show that open DQS can accurately reproduce the dynamics of open target systems prior to the threshold, but still exhibits a sudden performance breakdown beyond it. We introduce new, statistical techniques to rigorously characterise the onset of quantum chaos beyond the Trotterisation threshold for both approximately unitary and genuine dissipative quantum chaos, based on random matrix theory (RMT), building on techniques developed for closed systems. Finally, we demonstrate that dissipative quantum chaos emerges when the dissipation is in balance with the presence of underlying unitary quantum chaos; the dissipation must be strong enough to induce genuinely dissipative behaviour, yet not so strong as to suppress the underlying chaotic structure.