Quantum and Classical Dynamics with Random Permutation Circuits

TL;DR

This paper probes the role of quantumness in thermalisation by comparing minimally structured quantum dynamics (random unitary circuits) with their classical reversible counterparts (random permutation circuits). Using Weingarten calculus, the authors derive exact recurrences for averaged correlators, OTOCs, and purity, revealing that RPCs mimic the qualitative features of RUCs while introducing controlled quantitative differences. A parallel analysis in the classical setting shows decorrelators and mutual information obey the same light-cone–bounded spreading patterns as quantum OTOCs and entanglement growth, establishing deep formal correspondences between quantum and classical dynamics. The results imply a universal, mechanism-based similarity in operator spreading and information growth across quantum and classical locally interacting systems, while highlighting how basis constraints in RPCs yield measurable deviations in amplitudes and entanglement production. The work opens avenues for studying spectral properties, measurement-induced transitions, and practical implementation of pseudo-unitary designs using random permutations.

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.

Figures (2)

• Figure 1: Time evolution of an initial product state, represented by dark triangles (cf. Appendix \ref{['app:diagrammatics']} for a brief review of the diagrammatic notation).
• Figure 2: Rescaled correlation function $\expval{C_{\tau\tau}(x,t)}_{\rm RPC}/\Lambda_{\tau\tau}(t)$ (cf. Eq. \ref{['eq:avecorrresult']}), and its asymptotic form (cf. Eq. \ref{['eq:correlationasy']}) versus $x/\sqrt{t}$. The prefactor $\Lambda_{\tau\tau}(t)$ is chosen to remove the overall correlation decay, $\Lambda_{\mu\nu}(t)=K_{\mu\nu}(4/q)^{2t}(1+1/q)^{-4t}/t^2$. The label $\tau$ refers to the clock operator, $\mathcal{O}_{\tau}\ket{s}=\ket{(s+1)\pmod q}$, and we chose $q=5$.

Theorems & Definitions (1)

• Lemma 1