Probability Distributions and Weak Limit Theorems of Quaternionic Quantum Walks in One Dimension

Abstract

The discrete-time quantum walk (QW) is determined by a unitary matrix whose components are complex numbers. Konno (2015) extended the QW to the quaternionic quantum walk (QQW) whose components are quaternions and presented some properties of the QQW. Furthermore, Konno (2015) presented the question of whether or not the dynamics of a QQW is exactly the same as that of the corresponding QW. We give an answer to the problem by calculating the probability distribution and the weak limit density function of some classes of the QQW.