Pretty good state transfer in discrete-time quantum walks

TL;DR

This paper develops a theoretical framework for pretty good state transfer in discrete-time quantum walks, linking it to spectral properties of Hermitian adjacency matrices and providing constructions of walks exhibiting this phenomenon.

Contribution

It introduces a spectral characterization of pretty good state transfer in discrete-time quantum walks and constructs infinite families of such walks using advanced mathematical tools.

Findings

Pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrices.

Vertices involved must be m-strongly cospectral relative to this matrix.

Constructed infinite families of walks exhibiting pretty good state transfer.

Abstract

We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be $m$-strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.