# The spectra of the unitary marix of a 2-tessellable staggered quantum walk on a graph

## TL;DR

This paper derives explicit formulas for the spectra of the unitary matrices in 2-tessellable staggered quantum walks and Szegedy walks on graphs, with applications to quantum search algorithms.

## Contribution

It provides new spectral formulas for these quantum walk models, enhancing understanding of their dynamics and applications in quantum algorithms.

## Key findings

- Derived spectra formulas for 2-tessellable SQWs
- Provided spectra formulas for Szegedy matrices on bipartite graphs
- Applied results to quantum search characteristic polynomials

## Abstract

Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.