Limit theorems for discrete-time quantum walks on trees

We consider a discrete-time quantum walk Wt given by the Grover transformation on the Cayley tree. We reduce Wt to a quantum walk Xt on a half line with a wall at the origin. This paper presents two types of limit theorems for Xt. The first one is Xt as t → ∞, which corresponds to a localization in the case of an initial qubit state. The second one is Xt/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 Cayley tree.