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International Conference on Quantum Information

Quantum walks - types and properties

Igor Jex, M. Štefaňák, T. Kiss

Published Jan 1, 2008• 0 citations
Publisher

Abstract

We review properties of quantum walks with one and two excitations. The Polya number concept is applied to quantum walks and used for their classification. Quantum walks with random errors leading to localization are discussed.
DOI
10.1364/icqi.2008.qtuc4

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We analyze the recurrence probability (Pólya number) for d-dimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a by-product we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Pólya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.
Recurrence of quantum walks on lattices can be characterized by the generalized Pólya number. Its value reflects the difference between a classical and a quantum system. The dimension of the lattice is not a unique parameter in the quantum case; both the coin operator and the initial quantum state of the coin influence the recurrence in a nontrivial way. In addition, the definition of the Pólya number involves measurement of the system. Depending on how measurement is included in the definition, the recurrence properties vary. We show that in the limiting case of frequent, strong measurements, one can approach the classical dynamics. Comparing various cases, we have found numerical indication that our previous definition of the Pólya number provides an upper limit.
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.
Over the last two decades quantum walks have proven to be a very useful concept in quantum information processing, quantum communication and quantum simulations. The presented thesis reviews our contribution to this field of research. We provide a thorough introduction to methods used for the study of quantum walks and their potential applications. To keep the presentation intelligible we focus on specific examples. In particular, we analyze in detail the properties of homogeneous quantum walks on a one-dimensional lattice with two and three internal states. The application of the obtained results is illustrated on the examples persistence of unvisited sites and quantum transport to an absorbing sink. We also discuss quantum walks on finite graphs and their applications to quantum search and perfect state transfer. Extension of the presented results, which can be found in enclosed published papers, is discussed in detail.