Quantum Random Walk via Classical Random Walk With Internal States

In recent years quantum random walks have garnered much interest among quantum information researchers. Part of the reason is the prospect that many hard problems can be solved efficiently by employing algorithms based on quantum random walks, in the same way that classical random walks have played a central role in many hugely successful randomized algorithms. In this paper we introduce a new representation for the quantum random walks via the classical random walk with internal states. This new representation allows for a systematic approach to finding closed form expressions for the n-step distributions for a variety of quantum random walk models, and lends itself naturally to large deviation analysis. As an example, we show how to use the new representation to arrive at the same closed form expression for the Hadamard quantum random walk on a line, previously obtained by others. We assert the proposed method works in the most general settings.