Diffusive-ballistic crossover in 1D quantum walks

We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.