Random Walks: A Review of Algorithms and Applications

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 paper, 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.Comment: 13 pages, 4 figure

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa