Quantum Walks

Abstract

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.