Disordered Quantum Walks in one lattice dimension -

Andre Ahlbrecht, Volkher B. Scholz, Albert H. Werner - arXiv:1101.2298

We study a spin-$\frac{1}{2}$-particle moving on a one dimensional lattice
subject to disorder induced by a random, space-dependent quantum coin.
The discrete time evolution is given by a family of random unitary quantum
walk operators, where the shift operation is assumed to be deterministic.
Each coin is an independent identically distributed random variable with
values in the group of two dimensional unitary matrices. We derive sufficient
conditions on the probability distribution of the coins such that the system
exhibits dynamical localization. Put differently, the tunneling probability
between two lattice sites decays rapidly for almost all choices of random
coins and after arbitrary many time steps with increasing distance. Our
findings imply that this effect takes place if the coin is chosen at random
from the Haar measure, or some measure continuous with respect to it,
but also for a class of discrete probability measures which support consists
of two coins, one of them being the Hadamard coin.