Localization of quantum walks induced by recurrence properties of random walks

TL;DR

This paper investigates how the recurrence properties of a classical random walk influence the localization phenomena in a quantum walk derived from it, revealing fundamental links between classical recurrence and quantum localization.

Contribution

It establishes a direct connection between the recurrence characteristics of the classical random walk and the localization behavior of the associated quantum walk, including eigenvector and eigenvalue analysis.

Findings

Localization occurs when the RW is positively recurrent.

Eigenvectors in the  space are key to localization.

Presence of a single self loop impacts localization.

Abstract

We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization of the corresponding QW. We find the following two fundamental derivations of localization of the QW. The first one is the set of all the $\ell^2$ summable eigenvectors of the corresponding RW. The second one is the orthogonal complement, whose eigenvalues are $\pm 1$, of the subspace induced by the RW. In particular, as a consequence, for an infinite half line, we show that localization of the QW can be ensured by the positive recurrence of the corresponding RWs, and also that the existence of only one self loop affects localization properties.