Connection between continuous and discrete time quantum walks. From D-dimensional lattices to general graphs

I obtain the dynamics of the continuous time quantum walk on a d-dimensional lattice with periodic boundary conditions as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of [15] which proved this limit for the case of the quantum walk on the infinite line and the quantum cellular automaton proposed in [4]. By highlighting the main features of the limiting procedure, I then extend it to general regular graphs. For a given discrete time quantum walk on a graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics.