# Stationary measure for three-state quantum walk

## TL;DR

This paper derives the stationary measure for a one-dimensional three-state quantum walk using a new method based on the eigenvalue problem, providing explicit solutions and examples.

## Contribution

It introduces a novel recipe to obtain transfer matrices and solves the eigenvalue problem to find the stationary measure in general conditions.

## Key findings

- Derived the stationary measure for three-state quantum walk
- Provided explicit solutions for specific initial states
- Presented examples illustrating the stationary measure

## Abstract

We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.