Limit Theorems for Discrete-Time Quantum Walks on Trees

Abstract

We consider a discrete-time quantum walk W_t given by the Grover transformation on the homogeneous 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→∞, which corresponds to a localization in the case of an initial qubit state. The second one is X_t⁄t as t→∞, 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 homogeneous tree.