# Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model

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## Abstract

Coined quantum walks (QWs) are being used in many contexts with the goal of understanding quantum systems and building quantum algorithms for quantum computers. Alternative models such as Szegedy’s and continuous-time QWs were proposed taking advantage of the fact that quantum theory seems to allow different quantized versions based on the same classical model, in this case the classical random walk. In this work, we show the conditions upon which coined QWs are equivalent to Szegedy’s QWs. Those QW models have in common a large class of instances, in the sense that the evolution operators are equal when we convert the graph on which the coined QW takes place into a bipartite graph on which Szegedy’s QW takes place, and vice versa. We also show that the abstract search algorithm using the coined QW model can be cast into Szegedy’s searching framework using bipartite graphs with sinks.

## Keywords

Quantum walks Coined quantum walk Szegedy’s quantum walk Staggered quantum walk Equivalence among quantum walks## Notes

### Acknowledgments

The author acknowledges financial support from Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant Nos. 304709/2011-5, 474143/2013-9, and 400216/2014-0). The author thanks useful discussion with Raqueline A.M. Santos, Tharso D. Fernandes, Stefan Boettcher, Andris Ambainis, and the quantum computing group of LNCC.

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