Quantum random walks: an introductory overview

Abstract

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.Albert Einstein I. OVERVIEWEver 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 Peter Shor [4] discovered a quantum algorithm to factor numbers efficiently (that is in time that grows only polynomial with the length of the number to be factored), which 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 the physical devise that is carrying it ('Information is physical') has firmly settled among them and has led to fascinating new insights. Acquaintance with the basic notions of each of these fields seems instrumental in the understanding of modern quantum information processing.In this article we will follow the trajectory of one of the many surprising aspects of quantum information; it is dedicated to quantum random walks. We will give a thorough introduction to the necessary terminology without overburdening the reader with unnecessary mathematics. Starting with a very intuitive and physical example we will step by step introduce the language and notation of present day quantum information science. We will present the necessary background from computer science needed for a physicist to understand and appreciate the developments and results, assuming some rudimentary background of quantum mechanics, but no knowledge of computer science or quantum information theory. An excellent comprehensive introduction to quantum information and computation can be found in [1].In this journey at the interface of several traditional dis...

Two-dimensional Quantum Random Walk

“…In comparison to the classic symmetric random walk, the Hadamard quantum random walk (the symmetric version of the quantum random walk) propagates quadratically faster. The classic symmetric random walk after t steps has variance t, so the expected distance traveled is of order √ t. On the other hand, the Hadamard quantum random walk has a variance that scales with t 2 , so its expected distance traveled is of order t [Kem05].…”

“…Let p ∞ be the probability that the particle is absorbed, and then we have This theorem for the Hadamard QRW is again in sharp contrast to the symmetric CRW which has a probability of 1 of eventually exiting to the left. Furthermore, in the Hadamard QRW, if we change our initial starting point (or move the boundary) our absorption probability is still non-zero [Kem05].…”

Dynamical Localization of Quantum Walks in Random Environments

“…The dynamics of Quantum Walks (QW for short) have become a popular topic in the Quantum Computing community as the simplest quantum generalization of classical random walks, see for example the reviews [3], [21], [24]. In the same way classical random walks play an important role in theoretical computer science, typically in search algorithms, QW provide a natural and fruitful extension in the study of quantum search algorithms, see e.g.…”

Transitive State Alignment for the Quantum Jensen-Shannon Kernel

“…As a consequence, the evolution of the quantum walk is reversible, which implies that quantum walks are non-ergodic and do not possess a limiting distribution. See [9] for an overview of the properties of quantum walks. Using Dirac notation, we denote the basis state corresponding to the walk being at vertex u ∈ V as |u .…”