Controlling quantum walks with coin eigenstates

TL;DR

This paper demonstrates that expressing the initial state of a quantum walk in the coin operator's eigenbasis simplifies the analysis and enhances control over the walk's dynamics, providing deeper insights into its behavior.

Contribution

It introduces a method to analyze quantum walks using coin eigenstates, revealing extremal regimes and improving control over quantum walk dynamics.

Findings

Eigenbasis simplifies the group-velocity density expression

Initial coin state significantly influences quantum walk dynamics

Eigenvectors of the coin lead to extremal quantum walk regimes

Abstract

The control of quantum walk is made particularly transparent when the initial state is expressed in terms of the eigenstates of the coin operator. We show that the group-velocity density acquires a much simpler form when expressed in this basis. This allows us to obtain a much deeper understanding of the role of the initial coin state on the dynamics of quantum walks and control it. We find that the eigenvectors of the coin result in an extremal regime of a quantum walk. The approach is illustrated on two examples of quantum walks on a line.