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Linear Algebra and its Applications

The spectral analysis of the unitary matrix of a 2-tessellable staggered quantum walk on a graph

Norio Konno, Yusuke Ide, Iwao Sato

Published Feb 2, 2018• 2 citations
Publisher
DOI
10.1016/j.laa.2018.01.022

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Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its spectra. Furthermore, we present a formula for the Szegedy matrix of a bipartite graph by the same method, and so give its spectra. As an application, we present a formula for the characteristic polynomial of the modified Szegedy matrix in the quantum search problem on a graph, and give its spectra.
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.