Quantum graph walks I: mapping to quantum walks

TL;DR

This paper introduces quantum graph walks, a new class of coined quantum walks defined by quantum graphs, and shows their stationary states correspond to eigenfunctions of the quantum graph.

Contribution

It characterizes coined quantum walks via Euler circles and introduces quantum graph walks, linking quantum walks to quantum graph eigenfunctions.

Findings

Quantum graph walks are determined by specific quantum coins.

Stationary states of quantum graph walks correspond to quantum graph eigenfunctions.

The paper provides a new framework connecting quantum walks and quantum graphs.

Abstract

We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph.