# Quantum walks induced by Dirichlet random walks on infinite trees

## TL;DR

This paper analyzes the spectral properties of Grover quantum walks on infinite trees, revealing localization phenomena and connecting them to classical random walks with Dirichlet boundary conditions.

## Contribution

It provides a complete spectral characterization of the Grover walk on infinite trees, linking quantum localization to classical Dirichlet boundary conditions.

## Key findings

- Localization of the Grover walk on infinite trees
- Spectral characterization of the eigenspace
- Connection to classical random walks with Dirichlet boundary

## Abstract

We consider the Grover walk on infinite trees from the view point of spectral analysis. From the previous works, infinite regular trees provide localization. In this paper, we give the complete characterization of the eigenspace of this Grover walk, which involves localization of its behavior and recovers the previous works. Our result suggests that the Grover walk on infinite trees may be regarded as a limit of the quantum walk induced by the isotropic random walk with the Dirichlet boundary condition at the $n$-th depth rather than one with the Neumann boundary condition.