Localization of two-dimensional quantum walks defined by generalized Grover coins

TL;DR

This paper investigates localization phenomena in two-dimensional quantum walks with generalized Grover coins, revealing conditions under which localization occurs and how it depends on coin parameters.

Contribution

It introduces a class of parametric coin operators for 2D quantum walks and analyzes their localization properties, extending understanding beyond the standard Grover matrix.

Findings

Localization occurs at initial position for certain initial states with specific coin classes.

Localization depends on the coin parameter when the coin class does not include the Grover matrix.

The study characterizes conditions for localization in generalized quantum walks.

Abstract

Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this paper, however, the localization phenomena depends on the coin parameter when the class of parametric coins does not contain the Grover matrix.