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.
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.
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.
In the world of computational probing, random walks and quantum walks are two separate but tightly linked paradigms. Classical stochastic processes like random walks are based on the random movement of particles in mathematical space and have been extensively used in data mining, network analysis, computer vision, etc. Quantum walks, by contrast generalize random walks into the quantum realm, taking the benefit of the quantum superposition and interference in order to investigate multiple paths at the same time. Quantum walks provide significant potential for algorithmic development: in quantum computing and communication, and in Scientific modelling and simulation. This paper provides an in-depth survey of random walks and quantum walks, outlining historical context, fundamental principles, applications, challenges, and future prospects. In this paper, motivated by the abstractness of both random walks and quantum walks, we survey the field of computational exploration at large scales, coalescing the myriad notions and difficult challenges addressed by existing research efforts on random and quantum walks, inferring a nuanced and multifaceted view on how both random walks and quantum walks are involved in paving the way to future development in many fields.
Quantum computing is a developing field that applies the properties of quantum mechanics to develop solutions to complex problems that the most powerful classical computers cannot solve or cannot solve efficiently. Solving difficult tasks requires sophisticated and efficient algorithms that can utilize stochastic processes as the primary techniques. The typical models of these stochastic processes include discrete-time random walks and continuous ones. Quantum walks represent a quantum extension of discrete random walks, whose demonstrated utility in algorithm design has motivated signi cant interest in quantum computational research. Therefore, the purpose of this thesis is to explore the potential of discrete-time coined quantum walks using the Python package with high-performance computing (HPC) compared to classical walks by conducting a literature review. To supplement this research, some simulations are implemented to analyze the spreading behavior of coined quantum walks, compare them with classical walks, and demonstrate their application in search tasks, such as binary hypercube search algorithm. These simulations aim to clarify the extent to which quantum walks can be effectively modeled and applied in computational contexts. The results gained show that coined quantum walks exhibit faster spreading than classical random walks, which contributes to a deeper understanding of potential applications of quantum walks, such as quantum simulation, in the future.
A random walk is known as a random process which describes a path including a succession of random steps in the mathematical space. It has increasingly been popular in various disciplines such as mathematics and computer science. Furthermore, in quantum mechanics, quantum walks can be regarded as quantum analogues of classical random walks. Classical random walks and quantum walks can be used to calculate the proximity between nodes and extract the topology in the network. Various random walk related models can be applied in different fields, which is of great significance to downstream tasks such as link prediction, recommendation, computer vision, semi-supervised learning, and network embedding. In this article, we aim to provide a comprehensive review of classical random walks and quantum walks. We first review the knowledge of classical random walks and quantum walks, including basic concepts and some typical algorithms. We also compare the algorithms based on quantum walks and classical random walks from the perspective of time complexity. Then we introduce their applications in the field of computer science. Finally we discuss the open issues from the perspectives of efficiency, main-memory volume, and computing time of existing algorithms. This study aims to contribute to this growing area of research by exploring random walks and quantum walks together.
This paper investigates the application of quantum-inspired concepts to enhance the performance of random walk algorithms, specifically within the domain of global optimization. Classical random walk algorithms frequently struggle with convergence issues, frequently becoming trapped within local optima, a significant limitation when dealing with complex, high-dimensional search spaces. This research proposes a novel approach where random walk behavior is modeled based on principles derived from quantum mechanics, incorporating superposition and interference effects to facilitate more extensive and efficient exploration of the solution landscape. The core idea is to replace the deterministic step size in traditional random walks with a probabilistic movement governed by a quantum-like probability distribution. We explore the mathematical formulation of this process, detailing the probabilities of transition between states and the impact of interference. The resulting algorithms demonstrate improved convergence rates and a reduced tendency to get stuck in local optima compared to standard random walk methods. The paper concludes by outlining potential future research directions, including adaptive parameter tuning and integration with other optimization techniques.
Quantum computing aims to harness and exploit the quantum mechanical phenomena such as superposition, entanglement and contextuality in order to encode and process information. In this context, quantum walks, which has been suggested as the quantum counterpart of classical random walks, is an emerging topic in quantum computing that provides powerful techniques for developing new quantum algorithms, quantum simulation and quantum state transfer. This thesis intends to investigate the properties of quantum walks which may potentially promote further work in such techniques in quantum computation. We first propose a novel method for transferring arbitrary unknown qubit state between two points in space with quantum walk architecture. We determine the cases providing perfect state transfers over both finite and infinite lattices with different boundary conditions and we introduce recovery operators assisting the transfer process. Next, by modeling the incoherent and coherent transport with classical random walks and quantum walks, respectively, we calculate the transport efficiencies over an explosive percolation lattice. We show that the minimal correlation between discrete clusters leads to maximal localizations which originating from random scatterings. These localization effects, however, are rather small when compared to the supportive effect of the abrupt growth of the largest cluster on transport efficiency, which eventually allows us to obtain more efficient transports in case of minimal correlations. We support our results with further calculations on whether the eigenstates of the systems we study are localized or not. Lastly, we turn our attention to the spreading dynamics and the coin-position entanglement for two-dimensional quantum walks under an artificial magnetic fields by introducing Peierls phases to the system. In particular, we show that the spreading of the quantum walk is diffusive rather than ballistic when the ratio of the magnetic flux through unit cell - where the walk takes place - to the flux quantum is an irrational number. On the contrary, the walk recovers its original ballistic behavior when this ratio is chosen to be a rational number. Furthermore, we demonstrate that coin-position entanglement is nearly maximum under an artificial magnetic field, even for large number of steps.