Temporal Complexity Hierarchies in Solvable Quantum Many-Body Dynamics

TL;DR

This paper explores the computational complexity of the influence matrix in quantum many-body dynamics, revealing a hierarchy of complexity classes and distinguishing classical from quantum temporal correlations using group theory and tensor networks.

Contribution

It introduces a hierarchy of temporal entanglement scalings in quantum circuits and connects quantum dynamics complexity to geometric group theory, providing new insights into influence matrix properties.

Findings

Identified three distinct scalings of temporal entanglement entropy.

Established a hierarchy of computational resources for tensor-network representations.

Proposed an operational measure of quantum memory and demonstrated classical and quantum correlation distinctions.

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.