Propagation of correlations in Local Random Quantum Circuits

TL;DR

This paper establishes bounds on how correlations propagate in local random quantum circuits, revealing diffusive spreading and near-maximal correlations over time, with implications for quantum information dynamics.

Contribution

It provides a dynamical bound on correlation propagation in local random quantum circuits, characterizing spreading behavior and correlation saturation over time.

Findings

Correlations spread diffusively for times proportional to system size.

All parts of the system become nearly equally correlated for times less than quadratic in system size.

Correlations approach maximum asymptotically with exponentially suppressed corrections.

Abstract

We derive a dynamical bound on the propagation of correlations in local random quantum circuits - lattice spin systems where piecewise quantum operations - in space and time - occur with classical probabilities. Correlations are quantified by the Frobenius norm of the commutator of two positive operators acting on space-like separated local Hilbert spaces. For times $t=O(L)$ correlations spread to distances $\mathcal{D}=t$ growing, at best, diffusively for any distance within that radius with extensively suppressed distance dependent corrections whereas for $t=o(L^2)$ all parts of the system get almost equally correlated with exponentially suppressed distance dependent corrections and approach the maximum amount of correlations that may be established asymptotically.