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  • Open Access

Temporal Complexity Hierarchies in Solvable Quantum Many-Body Dynamics

He-Ran Wang1,*, Ilya Vilkoviskiy2,*, and Dmitry A. Abanin2,3,4,†

  • *These authors contributed equally to this work.
  • †Contact author: dabanin@princeton.edu
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Phys. Rev. X 16, 041041 – Published 13 November, 2026

DOI: https://doi.org/10.1103/3v7h-7dx9

Abstract

The influence matrix (IM) provides a powerful framework for characterizing nonequilibrium quantum many-body dynamics by encoding multitime correlations into tensor-network states. Understanding how its computational complexity relates to underlying dynamics is crucial for both theoretical insight and practical utility yet remains largely unexplored despite a few case studies. Here, we address this question for a family of brickwork quantum circuits ranging from integrable to chaotic regimes. Using tools from geometric group theory, we identify three qualitatively distinct scalings of temporal entanglement entropy, establishing a hierarchy of computational resources required for accurate tensor-network representations of the IM for these models. We further analyze the memory structure of the IM and distinguish between classical and quantum temporal correlations. In particular, for certain examples, we identify effectively classical IMs that admit an efficient Monte Carlo algorithm for computing multitime correlations. In more generic settings without an explicit classical description of the IM, we introduce an operational measure of quantum memory with an experimental protocol and discuss examples exhibiting long-time genuinely quantum correlations. Our results establish a new connection between quantum many-body dynamics and group theory, providing fresh insights into the complexity of the IM and its intricate connection to the physical characteristics of the dynamics.

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Popular Summary

Simulating out-of-equilibrium dynamics of interacting quantum particles is notoriously difficult, even when we care about the evolution of a small subsystem. The influence matrix describes how the rest of the system affects a subsystem over time, effectively storing the environmental quantum memory. How complex can this memory become? We construct exactly solvable influence matrices for a family of quantum models. By connecting their complexity to the growth of mathematical groups, we uncover several scenarios: The cost of influence-matrix representation can remain bounded, grow polynomially, or even grow exponentially with time. In some cases, this memory becomes purely classical, even for a genuinely quantum underlying system, allowing efficient predictions of local observables. In others, genuinely quantum memory persists for long times, and we propose a quantum teleportation protocol to detect it. Our findings reveal how memory shapes the complexity of quantum dynamics and suggest new strategies for efficient simulation.

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Outline

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Phys. Rev. X 16, 041041– Published 13 November, 2026
  • Received 14 November 2025
  • Revised 19 June 2026
  • Accepted 9 October 2026
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