The axioms of quantum physics imply that in general it makes no sense to speak of the long-term behaviour of a quantum walk. In this chapter we introduce a process that allows us to develop a meaningful substitute for a simple average.

To specify a discrete quantum walk on a graph, we need more than just the graph. In general we need some kind of ordering on the edges on each vertex, and this extra structure is closely related to machinery used to describe embeddings of graph in surfaces. in this chapter we explain this connection.

We present applications of the machinery developed in the previous chapter. The applications include examples of perfect state transfer, and a second treatment of Grover’s algorithm.

Aharonov et al. introduced class of quantum walks where the transition matrix is not (in general) a product of two reflections. (We call these shunt-decomposition walks.) In consequence, analysis of these walks is more difficult than the walks met with in earlier chapters. However the state space of walks is still the space of complex functions on the arcs of a graph. We give a description of these walks in graph theoretic terms, and study their behaviour.

We study shunt decomposition walks on the infinite path.

1. Grover Search: We introduce the basics of discrete quantum walks, describing some of the underlying physics. One of the most important algorithms in quantum computing is Grover’s search algorithm, we show how one can implement this algorithm using a discrete walk on the arcs of a graph.

We show how we can use embeddings of graphs in surfaces to contruct a new class of discrete walks and investigate their behaviour.

To analyse a discrete walk we need to compute the eigenvalues and eigenvectors of unitary matrix. The matrices that arise in practice are products of two reflections. We develop machinery that takes advantage of this structure to complete the specrtal information we need.