Quantum walks on graphs and quantum scattering theory

A number of classical algorithms are based on random walks on graphs. It is hoped that recently defined quantum walks can serve as the basis for quantum algorithms that will faster than the corresponding classical ones. We discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply advances one unit at each time step, so that its behavior there is analogous to free propatation. We are interested in how many steps it will take the particle, starting on one tail and propagating through the graph (where its propagation is not free), to emerge onto the other tail. The probability to make such a walk in n steps and the hitting time for such a walk can be expressed in terms of the transmission amplitude for the graph, which is one element of its S matrix. Demonstrating this neccessitates a study of the analyticity properties of the transmission and reflection coefficients of a graph. We show that a graph can have bound states that cannot be accessed by a particle entering the graph from one of the tails. Time-reversal invariance of a quantum walk is defined and used to show that the transmission amplitudes for the particle entering the graph from different directions are the same if the walk is time-reversal invariant.