Repeated measurements and random scattering in quantum walks

We study the effect of random scattering in quantum walks on a finite graph and 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 to design Hamiltonians whose eigenvectors are either localized or delocalized. By presenting some specific examples we demonstrate that the localization of eigenvectors restricts the transition probabilities on the graph and leads to a removal of energy states from the quantum walk in the monitored evolution. We conclude that repeated measurements as well as random scattering provide efficient tools for controlling quantum walks.