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Graph identification by quantum walks

Quantum random walks display remarkably different properties from their classical counterparts, most notably their fast spreading characteristics. For example, they were proven to provide an exponential algorithmic speedup for traversing a randomised glued-tree graph. However, despite such potentially superior efficiency in quantum random walks, they have yet to be applied to problems of practical importance. Graph isomorphism is a long- standing open problem in mathematics, which is to decide whether two given structures are topologically identical. This has applications in many areas of science and engineering. In this paper, we present an algorithm using quantum walks to solve the graph isomorphism problem. In particular, a novel measurement scheme is presented which makes it possible to identify graph isomorphism in polynomial time.