Continuous limits of linear and nonlinear quantum walks

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice delta Z uniformly converges (in Sobolev space H-s) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as delta -> 0. Here, to compare the walker defined on delta Z and the solution to the NLD defined on R, we use Shannon interpolation.