Directivity of quantum walk via its random walk replica

TL;DR

This paper explores the directivity of one-dimensional quantum walks by using their random walk replicas, revealing new insights into their spreading, localization, and return time properties through visualization and trajectory analysis.

Contribution

It introduces the use of quantum-walk-replicating random walks to analyze quantum walk directivity, offering novel visualization and understanding of their dynamics.

Findings

QWRWs allow trajectory-based analysis of QWs.

First return time differs significantly between QWs and RWs.

Visualization of transition probabilities reveals complex shapes.

Abstract

Quantum walks (QWs) exhibit different properties compared with classical random walks (RWs), most notably by linear spreading and localization. In the meantime, random walks that replicate quantum walks, which we refer to as quantum-walk-replicating random walks (QWRWs), have been studied in the literature where the eventual properties of QWRW coincide with those of QWs. However, we consider that the unique attributes of QWRWs have not been fully utilized in the former studies to obtain deeper or new insights into QWs. In this paper, we highlight the directivity of one-dimensional discrete quantum walks via QWRWs. By exploiting the fact that QWRW allows trajectories of individual walkers to be considered, we first discuss the determination of future directions of QWRWs, through which the effect of linear spreading and localization is manifested in another way. Furthermore, the transition probabilities of QWRWs can also be visualized and show a highly complex shape, representing QWs in a novel way. Moreover, we discuss the first return time to the origin between RWs and QWs, which is made possible via the notion of QWRWs. We observe that the first return time statistics of QWs are quite different from RWs, caused by both the linear spreading and localization properties of QWs.