This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent developments in this new area and outline some open questions. 1. Overview Ever since the discovery of quantum mechanics people have been puzzled by the counter-intuitive character of the laws of nature. Over time we have learned to accept more and more effects that are unimaginable in a classical Newtonian world. Modern technology exploits quantum effects both to our benefit and detriment—among the memorable examples we should cite laser technology and not omit the atomic bomb. In recent years interest in quantum information theory has been generated by the prospect of employing its laws to design devices of surprising power [1]. New ideas include quantum cryptography [2, 3] and quantum computation. In 1994 Shor [4] discovered a quantum algorithm to factor numbers efficiently (that is in time that grows only polynomically with the length of the number to be factored). This has unleashed a wave of activity across a broad range of disciplines: physics, computer science, mathematics and engineering. This fruitful axis of research has uncovered many new effects that are strikingly different from their classical counterparts, both from the physical point of view as well as from a computer science and communication theory perspective. Over time these communities have gained a greater understanding of the concepts and notions of the other. The idea that information cannot be separated from
This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent developments in this new area and outline some open questions. 1. Overview Ever since the discovery of quantum mechanics people have been puzzled by the counter-intuitive character of the laws of nature. Over time we have learned to accept more and more effects that are unimaginable in a classical Newtonian world. Modern technology exploits quantum effects both to our benefit and detriment—among the memorable examples we should cite laser technology and not omit the atomic bomb. In recent years interest in quantum information theory has been generated by the prospect of employing its laws to design devices of surprising power [1]. New ideas include quantum cryptography [2, 3] and quantum computation. In 1994 Shor [4] discovered a quantum algorithm to factor numbers efficiently (that is in time that grows only polynomically with the length of the number to be factored). This has unleashed a wave of activity across a broad range of disciplines: physics, computer science, mathematics and engineering. This fruitful axis of research has uncovered many new effects that are strikingly different from their classical counterparts, both from the physical point of view as well as from a computer science and communication theory perspective. Over time these communities have gained a greater understanding of the concepts and notions of the other. The idea that information cannot be separated from
This article aims to provide an introductory survey on quantum random walks.Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks.We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context.We will mention recent developments in this new area and outline some open questions. God does not play dice.
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.
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.
What are the information processing capabilities of physical systems?As recently as the first half of the 20 th century this question did not even have a definite meaning.What is information, and how would one process it?It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate.Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms?Rolf Landauer had the essential insight -"Information is physical" -that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics.This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave.Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences".Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]?One might begin to suspect foul play in the universe's operating principles.Interesting insights into the physics of information accumulated through the 1970s and 1980s -most sensationally in the proposal for a "quantum computer".If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer.But the applications of information in physics -and vice versa -have been far more widespread than this popular discovery.These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature.In this thesis I present a short review of ideas in quantum information theory.The first chapter contains introductory material, sketching the central ideas of probability and information theory.Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems.In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems.The final chapter presents a new resource: quantum information.This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.ii This thesis is dedicated to my mother.iii This thesis was begun as an intrepid adventure.It caught
Quantum physics is well known for being counter-intuitive, or even bizarre. Quantum correlations have no equivalence in classical physics. This was all well known for years. But several discoveries in the 1990's changed the world. First, in 1991 Artur Ekert, from Oxford University, discovered that quantum correlations could be used to distribute cryptographic keys. Suddenly physicists realized that quantum correlations and its associated bizarre non-locality, could be exploited to achieve a useful task that would be impossible without quantum physics. What a revolution! And this was not the end. Three years later, Peter Shor, from the AT&T Laboratories, discovered an algorithm that breaks the most used public key crypto-systems. Shor's algorithm requires a quantum computer, yet another bizarre quantum device, a kind of computer that heavily exploits the quantum superposition principle. The following year, in 1995, a collaboration between six physicists and computer scientists from three continents led to the discovery of quantum teleportation, a process with a science-fiction flavour. These discoveries and others led to the emergence of a new science, marrying quantum physics and theoretical computer science, called quantum information science. Today, a steadily growing community of physicists, mathematicians and computer scientists develop the tools of this new science. This leads to new experiments and new insights into physics and information theory. It is a fascinating time. Quantum information is still a very young science. In particular, the technology required to build a quantum computer is still unknown. Only quantum key distribution has reached a certain level of maturity, with a few start-ups already offering complete systems. Nevertheless, quantum information has been widely recognised as a source for truly innovative ideas and disruptive technologies.
Quantum Information Science is a cross-disciplinary subject that has arisen in the last twenty years. It concerns itself with the consequences of our most complete description of the physical world (that is, quantum mechanics) for the reliable, secure, private, and rapid processing of information, both in communication and computation. While its invention is often ascribed to the famous theoretical physicist Richard Feynman in the 1980s, his contributions were only one of many that initiated the field around that time. While he perceived that new types of physical devices, in which the quantum laws of superposition and entanglement function at the logical level, could give new power in the simulation of quantum physics, it was others (Bennett, Brassard) who showed that the uncertainty principle led to fundamentally more secure ways of communicating secret messages. It was yet others (Deutsch, Vazirani) who understood that quantum theory defined a new kind of Turing machine, and that not only quantum physics simulations, but potentially many other computational problems, are sped up in this machine. And it was yet another (Shor) who found a simple, very fast algorithm for prime factorization. The word "qubit" was coined only in 1995. Its introduction is indicative of a new mindset that has developed in recent years for studying and using quantum systems. "Qubit" now stands for a durable paradigm that spans a very wide variety of fields. It thus has many sub-meanings: first, it is the basic abstract unit of information for workers ranging from optical communication engineers to NMR spectroscopists to black-hole theorists. Second, it is the name we now routinely give to physical two level systems, as they are realized by photons, atomic and ionic eigenlevels, electronic and nuclear spin, structural defects in solids, and circulating-current states of superconducting devices. Physics has long dealt with some of these two-level systems, but calling them qubits implies a whole suite of interrelated ideas and capabilities that we ascribe to these systems: the ability to precisely set, and to precisely measure, the quantum state of one unique specimen, to avoid its coupling with the environment while at the same time strongly and controllably coupling the qubits together, to exploit the resulting quantum entanglement as a resource for metrological, cryptographic, and computational tasks. Quantum Information is a very diverse subject pursued today in many different directions, by many hundreds of researchers internationally: In theoretical physics, it has enlivened and sharpened the understanding of efficient representations of entangled many-particle wavefunctions, and has provided the prospect of applications for new concepts such as anyons and majorana fermions. Information theorists has benefitted from having a rigorous extension of the basis of their field, in which classical information theory is subsumed into a greatly broader subject. Theoretical computer scientists continue the search for new quantum algorithms, and have used quantum concepts to prove new results about the classification of computational complexity. Coding theorists have had the new and subtle problem of quantum error correction to analyse and conquer. Most strikingly, the program of experimental physics has been influenced in many directions by Quantum Information Science. State-of-the-art optics experiments transmit quantum states over long distances and perform precision manipulations in single quanta (atomic ions, impurity spins, quantum dots) in the quest to have working quantum cyptography and quantum computing. Quantum Hall systems, and topological insulators, are being assiduously examined for new elementary excitations for use as qubits. In the course of ten years, superconducting devices have improved by over four orders of magnitude in their quantum coherence, a metric made precise by the ideas of quantum computing. [...] DiVincenzo, David