Disordered Quantum Walks in one lattice dimension

TL;DR

This paper investigates how disorder in quantum coins affects the localization of a quantum walk on a one-dimensional lattice, demonstrating conditions under which the particle remains localized due to randomness.

Contribution

It provides new sufficient conditions on the distribution of random quantum coins that guarantee dynamical localization in one-dimensional quantum walks.

Findings

Dynamical localization occurs under certain probability distributions of the coins.

Localization is shown for Haar measure and some discrete measures including the Hadamard coin.

Tunneling probability decays rapidly with distance for almost all random coin choices.

Abstract

We study a spin-1/2-particle moving on a one dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.