Continuous deformations of the Grover walk preserving localization

TL;DR

This paper explores continuous deformations of the Grover walk on a line that preserve localization, revealing that localization is a property of entire families of coins rather than a single operator.

Contribution

It introduces two continuous deformations of the Grover walk that maintain localization, providing explicit formulas for peak velocities based on the coin parameter.

Findings

Localization persists across entire families of coin operators.

Peak velocities depend explicitly on the coin parameter.

Localization is not exclusive to a specific coin but a broader class.

Abstract

The three-state Grover walk on a line exhibits the localization effect characterized by a non-vanishing probability of the particle to stay at the origin. We present two continuous deformations of the Grover walk which preserve its localization nature. The resulting quantum walks differ in the rate at which they spread through the lattice. The velocities of the left and right-traveling probability peaks are given by the maximum of the group velocity. We find the explicit form of peak velocities in dependence on the coin parameter. Our results show that localization of the quantum walk is not a singular property of an isolated coin operator but can be found for entire families of coins.