  Quantum random walks are shown to have non-intuitive dynamics which makes
them an attractive area of study for devising quantum algorithms for
long-standing open problems as well as those arising in the field of quantum
computing. In the case of continuous-time quantum random walks, such peculiar
dynamics can arise from simple evolution operators closely resembling the
quantum free-wave propagator. We investigate the divergence of quantum walk
dynamics from the free-wave evolution and show that in order for
continuous-time quantum walks to display their characteristic propagation, the
state space must be discrete. This behavior rules out many continuous quantum
systems as possible candidates for implementing continuous-time quantum random
walks.
