# Classical simulation of quantum circuits by dynamical localization: analytic results for Pauli-observable scrambling in time-dependent disorder

## TL;DR

This paper demonstrates that certain quantum circuits exhibit dynamical localization, allowing classical simulation of quantum information scrambling through analytic formulas for out-of-time-ordered correlators in disordered spin systems.

## Contribution

It extends Anderson localization to quantum circuits, providing an analytic approach to simulate quantum dynamics efficiently in localized phases.

## Key findings

- Analytic formula for out-of-time-ordered correlator in matchgate circuits
- Efficient classical evaluation of local observable dynamics
- Identification of localized phases under disordered Hamiltonians

## Abstract

We extend the concept of Anderson localization, the confinement of quantum information in a spatially irregular potential, to quantum circuits. Considering matchgate circuits, generated by time-dependent spin-1/2 XY Hamiltonians, we give an analytic formula for the out-of-time-ordered correlator of a local observable, and show that it can be efficiently evaluated by a classical computer even when the explicit Heisenberg time evolution cannot. Because this quantity bounds the average error incurred by truncating the evolution to a spatially limited region, we demonstrate dynamical localization as a means for classically simulating quantum computation and give examples of localized phases under certain spatio-temporal disordered Hamiltonians.