# Stationary measure induced by the eigenvalue problem of the one-dimensional Hadamard walk

## TL;DR

This paper characterizes all stationary measures of the one-dimensional Hadamard quantum walk, classifying them into three types and providing conditions for periodicity of bounded measures using transfer matrix methods.

## Contribution

It completely determines stationary measures for the Hadamard walk and introduces a classification into quadratic polynomial, bounded, and exponential types.

## Key findings

- Stationary measures are classified into three types.
- Explicit conditions for periodicity of bounded measures are provided.
- Complete characterization of stationary measures via transfer matrix method.

## Abstract

In this paper, we consider the stationary measure of the Hadamard walk on the one-dimensional integer lattice. Here all the stationary measures given by solving the eigenvalue problem are completely determined via the transfer matrix method. Then these stationary measures can be divided into three classes, i.e., quadratic polynomial, bounded, and exponential types. In particular, we present an explicit necessary and sufficient condition for the bounded-type stationary measure to be periodic.