High state transfer in quantum walks on graphs

A quantum walk is a process that involves the evolution of a unit vector (i.e. of Euclidean norm 1), and is described by a unitary operator. The time-dependent distribution obtained by squaring the entries of the unit vector is quite different from that of a classical random walk. For instance, a property commonly occurring in random walks is convergence to a stationary distribution, something that does not generally occur in quantum walks. In this thesis, we study what we call 'state transfer' in various types of quantum walks on graphs (and other combinatorial structures), essentially trying to answer the following question: given a certain rule for constructing a unitary operator from a graph, for which graphs is there a high probability of ending up in a given vertex some time after the corresponding quantum walk has started at some initial vertex? We study properties called perfect state transfer (transfer with probability 1) and periodicity (transfer from a vertex to itself) in discrete-time quantum walks in Part I of this thesis. We also introduce a new notion called 'peak state transfer', for which the probability of transfer need not be 1. In Part II of the thesis, we bring this new notion to the continuous-time setting as well. As is the case for perfect state transfer and periodicity, the property of peak state transfer can be characterized in terms of the eigenvalues and eigenvectors of certain matrices that describe the underlying combinatorial structure.