Probability distributions and weak limit theorems of quaternionic quantum walks in one dimension

TL;DR

This paper investigates quaternionic quantum walks, extending traditional complex-based models, and establishes weak limit theorems to clarify their probability distribution differences from standard quantum walks.

Contribution

It provides the first weak limit theorems for quaternionic quantum walks, highlighting their distinct probabilistic behavior from classical quantum walks.

Findings

Probability distributions of QQWs differ from QWs.

Weak limit theorems characterize QQW behavior.

Numerical simulations support theoretical results.

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.