Scattering theory and discrete-time quantum walks

We study quantum walks on general graphs from the point of view of scattering theory. For a general finite graph we choose two vertices and attach one half line to each, and consider walks that proceed from one half line, through the graph, to the other. The probability of starting on one line and reaching the other after n steps can be expressed in terms of the transmission amplitude for the graph. Classical random walks on graphs can be used to construct algorithms that solve 2-SAT, graph connectivity problems, and for finding satsifying assignments for Boolean functions. A hope is that recently defined quantum walks will prove similarly useful in the development of quantum algorithms. In fact, there has been consderable recent progress is this regard. It has been shown that it is possible to use a quantum walk to perform a search on the hypercube faster than can be done classically [1]. In this problem the number of steps drops from N , which is the number of vertices, in the classical case to √ N in the quantum case. A much more dramatic improvement has recently been obtained by Childs, et al. [2]. They constructed an oracle problem that can be solved by a quantum algorithm based on a quantum walk exponentially faster than is possible with any classical algorithm. Quantum algorithms that are faster than any classical one have been found for for searching databases laid out in D dimensions using a continuous time walk [3] and in two dimensions using a discrete time walk [4]. Quantum-walk algorithms have also been found for element distinctness [5], finding triangles in graphs [6], subset finding [7], and determining whether a set of marked elements, which is promised to be of a certain size, exists or not [8].