Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

TL;DR

This paper establishes combinatorial necessary conditions for regular mixed graphs to induce periodic discrete-time quantum walks, linking eigenvalue properties to graph structure and providing criteria for specific classes like complete graphs.

Contribution

It introduces new combinatorial necessary conditions for periodicity in quantum walks on regular mixed graphs, connecting eigenvalue algebraic properties with graph coefficients.

Findings

If a k-regular mixed graph with n vertices is periodic, then 2n/k must be an integer.

Determined periodicity of mixed complete graphs.

Analyzed graphs with a prime number of vertices.

Abstract

We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. If the quantum walk is periodic, all the eigenvalues of the time evolution matrices must be algebraic integers. Focusing on this, we explore which ring the coefficients of the characteristic polynomials should belong to. On the other hand, the coefficients of the characteristic polynomials of $\eta$-Hermitian adjacency matrices have combinatorial implications. From these, we can find combinatorial implications in the coefficients of the characteristic polynomials of the time evolution matrices, and thus derive combinatorial necessary conditions for mixed graphs to be periodic. For example, if a $k$-regular mixed graph with $n$ vertices is periodic, then $2n/k$ must be an integer. As an application of this work, we determine periodicity of mixed complete graphs and mixed graphs with a prime number of vertices.