Two-step phantom relaxation of out-of-time-ordered correlations in random circuits

TL;DR

This paper investigates the relaxation dynamics of out-of-time-ordered correlations in random quantum circuits, revealing a two-step relaxation process governed by phantom and true eigenvalues of a propagator.

Contribution

It introduces a two-step relaxation framework for OTOCs in random circuits and derives exact dynamics and average OTOC expressions for finite systems.

Findings

Relaxation occurs in two distinct phases with different eigenvalue controls.

A phantom eigenvalue governs initial relaxation over long times.

Exact OTOC dynamics are obtained for circuits with two-qubit gates.

Abstract

We study out-of-time-ordered correlation (OTOC) functions in various random quantum circuits and show that the average dynamics is governed by a Markovian propagator. This is then used to study relaxation of OTOC to its long-time average value in circuits with random single-qubit unitaries, finding that relaxation in general proceeds in two steps: in the first phase that lasts upto an extensively long time the relaxation rate is given by a phantom eigenvalue of a non-symmetric propagator, whereas in the second phase the rate is determined by the true 2nd largest propagator eigenvalue. We also obtain exact OTOC dynamics on the light-cone and an expression for the average OTOC in finite random circuits with random two-qubit gates.