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arXiv:1811.02116 · Linear Algebra and its Applications

Eigenbasis of the evolution operator of 2-tessellable quantum walks

Yusuke Higuchi, Renato Portugal, Iwao Sato, Etsuo Segawa

Published Aug 27, 2019• 6 citations
arXivHTMLPDF
DOI
10.1016/j.laa.2019.08.019

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A discrete-time staggered quantum walk was recently introduced as a generalization that allows the unification of other versions, such as the coined and Szegedy's walk. However, it also produces forms of quantum walks not covered by previous versions. To explore their properties, we study here the staggered walk on a hexagonal lattice. Such a walk is defined using a set of overlapping tessellations that cover the graph edges, and each tessellation is a partition of the node set into cliques. The hexagonal lattice requires at least three tessellations. Each tessellation is associated with a local unitary operator and the product of the local operators defines the evolution operator of the staggered walk on the graph. After defining the evolution operator on the hexagonal lattice, we analyze the quantum walk dynamics with the focus on the position standard deviation and localization. We also obtain analytic results for the time complexity of spatial search algorithms with one marked node using cyclic boundary conditions.
A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely studied coined walk. Coined walks require the direct product of the site basis with the coin space; coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.