Recurrences in three-state quantum walks on a plane

Abstract

We analyze the role of dimensionality in the time evolution of discrete time quantum walks through the example of the three-state walk on a two-dimensional, triangular lattice. We show that the three-state Grover walk does not lead to trapping (localization) or recurrence to the origin, in sharp contrast to the Grover walk on the two dimensional square lattice. We determine the power law scaling of the probability at the origin with the method of stationary phase. We prove that only a special subclass of coin operators can lead to recurrence and there are no coins leading to localization. The propagation for the recurrent subclass of coins is quasi one-dimensional.

7 pages, 4 figures

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• Transport properties of continuous-time quantum walks on Sierpinski fractals
• Directional correlations in quantum walks with two particles
• Continuous deformations of the Grover walk preserving localization
• Localization in Quantum Walks on a Honeycomb Network
• Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models
• Complete classification of trapping coins for quantum walks on the 2D square lattice
• Limit theorems and localization of three state quantum walks on a line defined by generalized Grover coins
• Spectral and asymptotic properties of Grover walks on crystal lattice
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• Bifurcation Curves of Two-Dimensional Quantum Walks