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Journal of Advances and Scholarly Researches in Allied Education

A Research on some Approaches and Properties of Quantum Graphs: an Introduction

Mithleshwari Ghormare, Alok Kumar

Published Jul 1, 2018• 0 citations
Publisher
DOI
10.29070/15/57893

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Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.

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Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
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.