Limit distributions of two-dimensional quantum walks
Physical Review A—Atomic, Molecular, and Optical Physics, 2008•APS
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→∞ 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 …
includes the Grover-walk model as a special case, is analytically studied. Convergence in
the long-time limit t→∞ 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 …
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 of all joint moments of two components of walker’s pseudovelocity, and , 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.
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