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arXiv:0802.2749 · Physical Review A

Limit distributions of two-dimensional quantum walks

恭平 渡部, Naoki Kobayashi, Makoto Katori, Norio Konno

Published Jun 19, 2008• 91 citations
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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.

DOI
10.1103/physreva.77.062331

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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→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞ of all joint moments of two components of walker's pseudovelocity, Xt/t{X}_{t}/tXt​/t and Yt/t{Y}_{t}/tYt​/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.
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