The staggered quantum walk model
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There are at least three models of discrete-time quantum walks (QWs) on graphs currently under active development. In this work, we focus on the equivalence of two of them, known as Szegedy’s and staggered QWs. We give a formal definition of the staggered model and discuss generalized versions for searching marked vertices. Using this formal definition, we prove that any instance of Szegedy’s model is equivalent to an instance of the staggered model. On the other hand, we show that there are instances of the staggered model that cannot be cast into Szegedy’s framework. Our analysis also works when there are marked vertices. We show that Szegedy’s spatial search algorithms can be converted into search algorithms in staggered QWs. We take advantage of the similarity of those models to define the quantum hitting time in the staggered model and to describe a method to calculate the eigenvalues and eigenvectors of the evolution operator of staggered QWs.
KeywordsQuantum walks Staggered model Coinless model Szegedy Hitting time
RP acknowledges financial support from Faperj (Grant No. E-26/102.350/2013) and CNPq (Grants Nos. 304709/2011-5, 474143/2013-9, and 400216/2014-0). RAMS acknowledges financial support from Capes/Faperj E-45/2013. RP thanks helpful discussions with Stefan Boettcher and Andris Ambainis’ group. The authors thank the anonymous referees for useful suggestions.
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