Limit distributions of two-dimensional quantum walks

Abstract

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit $t\ensuremath{\rightarrow}\ensuremath{\infty}$ of all joint moments of two components of walker's pseudovelocity, ${X}_{t}/t$ and ${Y}_{t}/t$, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.