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AIP conference proceedings

Quantum walk and potential application

Jingbo B. Wang, B. L. Douglas, John Gideon Hartnett, Paul C Abbott

Published Jan 1, 2010• 2 citations
Publisher

Abstract

Quantum walk represents a generalised version of the well‐known classical random walk. Regardless of their apparent connection, the dynamics of quantum walk is often non‐intuitive and far deviate from its classical counterpart. However, despite such potentially superior efficiency in quantum walks, it has yet to be applied to problems of practical importance. In this paper, we will give a brief introduction to quantum walks and discuss potential applications.

DOI
10.1063/1.3460202

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Quantum walk represents a generalised version of the well‐known classical random walk. Regardless of their apparent connection, the dynamics of quantum walk is often non‐intuitive and far deviate from its classical counterpart. However, despite such potentially superior efficiency in quantum walks, it has yet to be applied to problems of practical importance. In this paper, we will give a brief introduction to quantum walks and discuss potential applications.
The classical random walk formalism plays an important role in a wide range of applications. Its quantum counterpart, the quantum walk, is proposed as an important theoretical model for quantum computing. By exploiting quantum effects such as superposition, interference, and entanglement, quantum walks and their variations have been extensively studied for achieving computing power beyond that of classical computing and have been broadly used in designing quantum algorithms for algebraic and optimization problems, graph and network analysis, and quantum Hamiltonian and biochemical process simulations. Moreover, quantum walk models have been proven capable of universal quantum computation. Unlike conventional quantum circuit models, quantum walks provide a feasible path for implementing application-specific quantum computing, particularly in the noisy intermediate-scale quantum era. Recently, remarkable progress has been achieved in implementing a wide variety of quantum walks and quantum walk applications, which demonstrates the great potential of quantum walks. In this review, we provide a thorough summary of quantum walks and quantum walk computing, including theories and characteristics, physical implementations, and applications. We also discuss the challenges facing quantum walk computing, which aims to realize a practical quantum computer in the near future.
Abstract Quantum walk is the quantum version of classical random walk, which has shown great advantage over classical algorithms. In this article, main discoveries and breakthroughs along with several applications in the last two decades are reviewed and discussed. Reducing hitting time and diminishing mixing time, as two hot research fields, are addressed. Possibility of universal computation with more generic case of quantum walk is also mentioned. In the future, we can expect new methods and models being feasible ways to approach the theoretical lower bounds of related problems, as well as more efficient applications on specific scene.
In this thesis we present a theoretical study of quantum walks, with a particular focus on the development of viable schemes concerned with their physical realization. Ever since their introduction over a decade ago, quantum walks have been extensively explored for their non-intuitive dynamics which may hold the key to a new generation of quantum algorithms. This growing interest in the theoretical applications of quantum walks has been paralleled by a flurry of research into a more practical problem: how does one physically implement a quantum walk in the laboratory? We begin this thesis by first presenting an overview of the quantum walk theory, including some of its algorithmic applications. This is then followed by a comprehensive survey of numerous proposals for a physical implementation of quantum walks, underpinned by a wide range of quantum, classical and hybrid technologies. This review consequently highlights what has so far remained a major challenge for the quantum walk enthusiasts; a physical realization that is experimentally viable whilst remaining readily scalable and not limited to problems with specific connectivity criteria. It is precisely this challenge that we seek to examine in the remaining parts of this thesis.
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.
The development of quantum algorithms based on quantum versions of random walks is placed in the context of the emerging field of quantum computing. Constructing a suitable quantum version of a random walk is not trivial; pure quantum dynamics is deterministic, so randomness only enters during the measurement phase, i.e. when converting the quantum information into classical information. The outcome of a quantum random walk is very different from the corresponding classical random walk owing to the interference between the different possible paths. The upshot is that quantum walkers find themselves further from their starting point than a classical walker on average, and this forms the basis of a quantum speed up, which can be exploited to solve problems faster. Surprisingly, the effect of making the walk slightly less than perfectly quantum can optimize the properties of the quantum walk for algorithmic applications. Looking to the future, even with a small quantum computer available, the development of quantum walk algorithms might proceed more rapidly than it has, especially for solving real problems.