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 component is complex number. Konno (2015) extended the QW to a walk whose component is quaternion.We call this model quaternionic quantum walk (QQW). The probability distribution of a class of QQWs is the same as that of the QW. On the other hand, a numerical simulation suggests that the probability distribution of a QQW is different from the QW. In this paper, we clarify the difference between the QQW and the QW by weak limit theorems for a class of QQWs. Definitons and models 2.1 QuaternionThe quaternion was introduced as an extension of the complex number by Hamilton in 1843. Let R, C and H be the set of real, complex numbers and quaternions, respectively. Then x ∈ H is expressed as follows.x = x 0 + x 1 i + x 2 j + x 3 k ∈ H, * saito-kei-nb@ynu.jp Abbr. title: Probability distributions and weak limit theorems of quaternionic quantum walks in one dimension AMS 2000 subject classifications: 60F05, 81P68

A new time-series model based on quantum walk

“…Recently the author extended the QW to a new walk called quaternionic quantum walk (QQW) determined by a unitary matrix whose component is quaternion [13]. In general, the behavior of QQW is different from usual QW [15,24], it is interesting to compare our method with a time-series one based on the QQW model. Moreover an extension from the QQW time-series model to the Clifford algebra time-series one would be also attractive.…”