Repeated measurements and random scattering in quantum walks

Abstract We study the effect of random scattering in quantum walks on a finite graphand compare it with the effect of repeated measurements. To this end, a constructive approach is employed by introducing a localized and a delocalized basis for the underlying Hilbert space. This enables us todesign Hamiltonians whose eigenvectors are either localized or delocalized.By presenting some specific examples we demonstrate that the localization ofeigenvectors restricts the transition probabilities on the graph and leads toa removal of energy states from the quantum walk in the monitored evolution. We conclude that repeated measurementsas well as random scattering provide efficient tools for controlling quantum walks.