A Classical approach to the graph isomorphism problem using quantum walks

Given the extensive application of classical random walks to classicalalgorithms in a variety of fields, their quantum analogue in quantum walksis expected to provide a fruitful source of quantum algorithms. So far,however, such algorithms have been scarce. In this work, we enumeratesome important differences between quantum and classical walks, leadingto their markedly different properties. We show that for many practicalpurposes, the implementation of quantum walks can be efficiently achievedusing a classical computer. We then develop both classical and quantum graphisomorphism algorithms based on discrete-time quantum walks. We showthat they are effective in identifying isomorphism classes of large databasesof graphs, in particular groups of strongly regular graphs. We consider thisapproach to represent a promising candidate for an efficient solution to the graphisomorphism problem, and believe that similar methods employing quantumwalks, or derivatives of these walks, may prove beneficial in constructing otheralgorithms for a variety of purposes.