# A spectral analysis of discrete-time quantum walks related to the birth and death chains

## TL;DR

This paper performs a spectral analysis of discrete-time quantum walks on paths, linking their distributions to eigenvalues and eigenvectors of associated birth and death chains, with applications to Szegedy's walk and the Ehrenfest model.

## Contribution

It establishes a connection between quantum walk distributions and spectral properties of birth and death chains, providing explicit formulas for specific models.

## Key findings

- Time-averaged distribution expressed via eigenvalues and eigenvectors.
- Connection between quantum walks and classical birth-death chains.
- Application to Szegedy's walk and derivation of the arcsine law.

## Abstract

In this paper, we consider a spectral analysis of discrete time quantum walks on the path. For isospectral coin cases, we show that the time averaged distribution and stationary distributions of the quantum walks are described by the pair of eigenvalues of the coins as well as the eigenvalues and eigenvectors of the corresponding random walks which are usually referred as the birth and death chains. As an example of the results, we derive the time averaged distribution of so-called Szegedy's walk which is related to the Ehrenfest model. It is represented by Krawtchouk polynomials which is the eigenvectors of the model and includes the arcsine law.