Quantum walks with random phase shifts
  We investigate quantum walks in multiple dimensions with different quantum coins. We augment the model by assuming that at each step the amplitudes of the coin state are multiplied by random phases. This model enables us to study in detail the role of decoherence in quantum walks and to investigate the quantum-to-classical transition. We also provide classical analogues of the quantum random walks studied. Interestingly enough, it turns out that the classical counterparts of some quantum random walks are classical random walks with a memory and biased coin. In addition random phase shifts "simplify" the dynamics (the cross interference terms of different paths vanish on average) and enable us to give a compact formula for the dispersion of such walks. 
Jozef Kosik, Vladimir Buzek, Mark Hillery