# Eigenvalues of quantum walks of Grover and Fourier types

## TL;DR

This paper characterizes eigenvalues of certain quantum walks, showing localization phenomena for Grover walks and absence of localization for Fourier walks, with implications for quantum walk behavior and structure.

## Contribution

It provides necessary and sufficient conditions for eigenvalues in periodic quantum walks and applies these to analyze Grover and Fourier type walks.

## Key findings

- Grover walks in any dimension have eigenvalues at ±1.
- Lazy Grover walks have eigenvalue 1, causing localization.
- Two-dimensional Fourier walk has no eigenvalues, thus no localization.

## Abstract

A necessary and sufficient conditions for certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of $\pm 1$ as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has $1$ as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general criterion for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics such as Grover walks on the triangular lattice, products and deformations of Grover walks are also discussed.