Quantum Walk on the Line

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider quantum walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the “Hadamard walk”. In particular, we show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [−t/ √ 2, t/ √ 2]. This implies that the same walk defined on the circle mixes in linear time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.