Initial State Memory in Finite Random Brickwork Circuits

TL;DR

This paper investigates how finite random brickwork circuits retain or lose local information about initial states, revealing a size-dependent memory retention, universal dynamics, and a phase transition induced by boundary dissipation.

Contribution

It provides an exact characterization of information retention in finite brickwork circuits and identifies conditions for memory preservation and phase transitions.

Findings

Memory retained if environment is smaller than half the system

Frobenius distance becomes universal at large scales

Boundary dissipation induces a phase transition

Abstract

We ask under what conditions a finite brickwork circuit of random gates retains local information about the initial state. To answer this question we measure the averaged Frobenius distance between the reduced states obtained by evolving two arbitrary initial states and tracing out a portion of the system. By characterising this distance exactly at all times we find that the information is retained if the environment -- the subsystem traced out -- is smaller than half of the system and washed away otherwise. We also find that, while the dynamics of the Frobenius distance depends on the specific initial states chosen, this dependence becomes increasingly weak for large scales and eventually the Frobenius distance attains a universal form as a function of time. Finally, we show that by introducing weak enough boundary dissipation, one can observe a phase transition between a memory preserving phase and one where the information is completely lost.