Quantum and Classical Dynamics with Random Permutation Circuits

TL;DR

This paper investigates the similarities and differences between quantum and classical thermalisation processes using random permutation circuits and compares their dynamics through analytical calculations of key quantities.

Contribution

It introduces random permutation circuits as a classical analogue to quantum circuits and establishes exact relations between quantum and classical dynamical quantities.

Findings

Quantum and classical dynamics show qualitatively similar behaviours.

Exact relations connect quantum purity with classical mutual information growth.

RPCs and RUCs allow analytical computation of OTOCs and entanglement entropies.

Abstract

Understanding thermalisation in quantum many-body systems is among the most enduring problems in modern physics. A particularly interesting question concerns the role played by quantum mechanics in this process, i.e. whether thermalisation in quantum many-body systems is fundamentally different from that in classical many-body systems and, if so, which of its features are genuinely quantum. Here we study this question in minimally structured many-body systems which are only constrained to have local interactions, i.e. local random circuits. We introduce a class of random permutation circuits (RPCs), where the gates locally permute basis states modelling generic microscopic classical dynamics, and compare them to random unitary circuits (RUCs), a standard toy model for generic quantum dynamics. We show that, like RUCs, RPCs permit the analytical computation of several key quantities such as out-of-time order correlators (OTOCs), or entanglement entropies. RPCs can be interpreted both as quantum or classical dynamics, which we use to find similarities and differences between the two. Performing the average over all random circuits, we discover a series of exact relations, connecting quantities in RUC and (quantum) RPCs. In the classical setting, we obtain similar exact results relating (quantum) purity to (classical) growth of mutual information and (quantum) OTOCs to (classical) decorrelators. Our results indicate that despite of the fundamental differences between quantum and classical systems, their dynamics exhibits qualitatively similar behaviours.