Unitary equivalent classes of one-dimensional quantum walks

TL;DR

This paper classifies one-dimensional quantum walks into unitary equivalent classes, showing their relation to Ambainis type and Szegedy walks, and provides conditions for such classifications.

Contribution

It establishes the unitary equivalence of one-dimensional quantum walks to Ambainis type and Szegedy walks, offering a complete characterization.

Findings

One-dimensional quantum walks are unitary equivalent to Ambainis type walks.

Translation-invariant quantum walks are Szegedy walks.

A necessary and sufficient condition for a walk to be a Szegedy walk is provided.

Abstract

This study investigates unitary equivalent classes of one-dimensional quantum walks. We prove that one-dimensional quantum walks are unitary equivalent to quantum walks of Ambainis type and that translation-invariant one-dimensional quantum walks are Szegedy walks. We also present a necessary and sufficient condition for a one-dimensional quantum walk to be a Szegedy walk.