A note on the spectral mapping theorem of quantum walk models

TL;DR

This paper explores the spectral properties of quantum walk models, establishing a spectral mapping theorem that relates the eigenspaces of the walk operator to those of an associated linear operator without requiring spectral decomposition.

Contribution

It provides a new spectral mapping theorem for quantum walks that does not depend on the spectral decomposition of the associated operator T.

Findings

Eigenspaces of quantum walk operators characterized by generalized kernels of T

Spectral mapping theorem established without spectral decomposition of T

Applicable to Szegedy walk on graphs

Abstract

We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of $U$ without the spectral decomposition of $T$. Arguments in this direction reveal the eigenspaces of $U$ characterized by the generalized kernels of linear operators given by $T$.