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 $t\to \infty$ 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.

v3: REVTeX4, 20 pages, 6 figures, minor corrections made for publication

References in corpus (3)

• Spatial search by quantum walk
• Spatial search and the Dirac equation
• Quantum walks and orbital states of a Weyl particle