Exponentially Decaying Velocity Bounds of Quantum Walks in Periodic Fields

We consider a class of discrete-time one-dimensional quantum walks, associated with CMV unitary matrices, in the presence of a local field. This class is parametrized by a transmission parameter t∈ [0 , 1] . We show that for a certain range for t, the corresponding asymptotic velocity can be made arbitrarily small by introducing a periodic local field with a sufficiently large period. In particular, we prove an upper bound for the velocity of the n-periodic quantum walk that is decaying exponentially in the period length n. Hence, localization-like effects are observed even after a long number of quantum walk steps when n is large.