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arXiv · 0802.2749

Limit distributions of two-dimensional quantum walks

Kyohei Watabe·Naoki Kobayashi·Makoto Katori·Norio Konno
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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 $t \to \infty$ 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.

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Kyohei Watabe, Naoki Kobayashi, Makoto Katori, Norio Konno. 2008-06-20. Limit distributions of two-dimensional quantum walks. https://doi.org/10.1103/physreva.77.062331

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