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arXiv:2201.08067 · Complexity

Directivity of Quantum Walk via Its Random Walk Replica

Tomoki Yamagami, Etsuo Segawa, Nicolas Chauvet, André Röhm, Ryoichi Horisaki, Makoto Naruse

Published Jan 1, 2022• 2 citations
arXivHTMLPDF

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.

DOI
10.1155/2022/9021583

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Quantum computers have sparked significant interest in recent years due to their potential to revolutionize various fields. One area where quantum computing shows great promise is in the study of quantum walks, a quantum counterpart of the classical random walk algorithm that has been foundational in scientific research. While both quantum and classical random walks involve a "walker" moving through a space or graph, quantum walks differ fundamentally due to quantum principles such as superposition, leading to unique behaviors like linear spreading and localization. This thesis investigates the quantum walk simulation, with a particular focus on the Quantum-Walk-Replicating-Random-Walk (QWRW) model. Unlike traditional quantum walks, which sum over all possible paths with complex interference effects, the QWRW approach models the walk as a series of distinct, classical-like steps. This trajectory-based perspective offers a novel way to analyze the walker's position and movement, avoiding the complexities of quantum interference. The QWRW model is particularly valuable in understanding key phenomena of quantum walks, such as linear spreading and localization, by providing insights into the directional properties of quantum walkers. By defining transition probabilities in both space and time, the QWRW model offers a detailed framework for examining the spatial and temporal characteristics of quantum walks, enhancing our understanding of their behavior and potential applications. This study aims to bridge the gap between classical and quantum walks, contributing to the broader field of quantum computing and its practical implications.
We study a quantum walk (QW) whose time evolution is induced by a random walk (RW) first introduced by Szegedy (2004). We focus on a relation between recurrent properties of the RW and localization of the corresponding QW. We find the following two fundamental derivations of localization of the QW. The first one is the set of all the ℓ2\ell^2ℓ2 summable eigenvectors of the corresponding RW. The second one is the orthogonal complement, whose eigenvalues are ±1\pm 1±1, of the subspace induced by the RW. In particular, as a consequence, for an infinite half line, we show that localization of the QW can be ensured by the positive recurrence of the corresponding RWs, and also that the existence of only one self loop affects localization properties.
In this paper we claim that a common underlying structure---a skeleton structure---is present behind discrete-time quantum walks (QWs) on a one-dimensional lattice with a homogeneous coin matrix. This skeleton structure is independent of the initial state, and partially, even of the coin matrix. This structure is best interpreted in the context of quantum-walk-replicating random walks (QWRWs), i.e., random walks that replicate the probability distribution of quantum walks, where this structure acts as a simplified formula for the transition probability. Additionally, we construct a random walk whose transition probabilities are defined by the skeleton structure and demonstrate that the resultant properties of the walkers are similar to both the original QWs and QWRWs.
The discrete-time quantum walk (QW) is a quantum version of the random walk (RW) and has been widely investigated for the last two decades. Some remarkable properties of QW are well known. For example, QW has a ballistic spreading, i.e., QW is quadratically faster than RW. For some cases, localization occurs: a walker stays at the starting position forever. In this paper, we consider stationary measures of two-state QWs on the line. It was shown that for any space-homogeneous model, the uniform measure becomes the stationary measure. However, the corresponding result for space-inhomogeneous model is not known. Here, we present a class of space-inhomogeneous QWs on the line and cycles in which the uniform measure is stationary. Furthermore, we briefly discuss a difference between QWs and RWs.
Abstract In this paper, we study the properties of lackadaisical quantum walks (LQWs) on a line. This model was first proposed in Wong (2015 J. Phys. A: Math. Theor . 48 435304) as a quantum analogue of lazy random walks where each vertex is attached to τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate the wave function of the walker starting from the origin and obtain a long-time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution for the variance of the walker’s probability distribution.