Limit theorems for discrete-time quantum walks on trees

TL;DR

This paper studies the long-term behavior of discrete-time quantum walks on Cayley trees, establishing limit theorems that describe localization and asymptotic distribution, with connections to known density functions.

Contribution

It introduces limit theorems for quantum walks on trees by reducing the problem to walks on a half line, extending understanding of their asymptotic behavior.

Findings

Localization occurs for certain initial states.

The scaled walk converges to a distribution given by the Konno density.

Results connect discrete-time quantum walks on trees to known density functions.

Abstract

We consider a discrete-time quantum walk W_t given by the Grover transformation on the Cayley tree. We reduce W_t to a quantum walk X_t on a half line with a wall at the origin. This paper presents two types of limit theorems for X_t. The first one is X_t as t\to\infty, which corresponds to a localization in the case of an initial qubit state. The second one is X_t/t as t\to\infty, whose limit density is given by the Konno density function [1-4]. The density appears in various situations of discrete-time cases. The corresponding similar limit theorem was proved in [5] for a continuous-time case on the Cayley tree.