Abstract
The staggered quantum walk (SQW) model is defined by partitioning the graph into cliques, which are called polygons. We analyze the role that the size of the polygon intersection plays on the dynamics of SQWs on graphs. We introduce two processes (intersection reduction and intersection expansion), that change the number of vertices in some intersection of polygons, and we compare the behavior of the SQW on the reduced or expanded graph in relation to the SQW on the original graph. We describe how the eigenvectors and eigenvalues of the evolution operators relate to each other. This processes can help to establish the equivalence between SQWs on different graphs and to simplify the analysis of SQWs. We also show an example of a SQW on a graph that is not included in Szegedy’s model, but which is equivalent to an instance of Szegedy’s model after applying the intersection reduction.
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References
Abreu A, Cunha L, Fernandes T, de Figueiredo C, Kowada L, Marquezino F, Posner D, Portugal R (2018) The graph tessellation cover number: extremal bounds, efficient algorithms and hardness. In: Bender MA, Farach-Colton M, Mosteiro MA (eds) LATIN 2018: theoretical informatics. Springer, Cham, pp 1–13
Aharonov Y, Davidovich L, Zagury N (1993) Quantum random walks. Phys Rev A 48(2):1687–1690
Aharonov D, Ambainis A, Kempe J, Vazirani U (2001) Quantum walks on graphs. In: Proceedings of the thirty-third annual ACM symposium on theory of computing, STOC ’01, New York, pp 50–59
Ambainis A (2004) Quantum walk algorithm for element distinctness. In: Proceedings of the 45th annual IEEE symposium on foundations of computer science
Ambainis A, Kempe J, Rivosh A (2005) Coins make quantum walks faster. In: Proceedings of the 16th ACM-SIAM symposium on discrete algorithms, pp 1099–1108
Chagas BA, Portugal R, Boettcher S, Segawa E (2018) Staggered quantum walk on hexagonal lattices. arXiv:quant-ph:1806.10249
Coutinho G, Portugal R (2018) Discretization of continuous-time quantum walks via the staggered model with hamiltonians. Nat Comput 18(2):403–409
Farhi E, Gutmann S (1998) Quantum computation and decision trees. Phys Rev A 58:915–928
Moqadam JK, de Oliveira MC, Portugal R (2017) Staggered quantum walks with superconducting microwave resonators. Phys Rev B 95:144506
Portugal R (2016a) Staggered quantum walks on graphs. Phys Rev A 93:062335
Portugal R (2016b) Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model. Quantum Inf Process 15(4):1387–1409
Portugal R (2018) Element distinctness revisited. Quantum Inf Process 17(7):163
Portugal R, Fernandes TD (2017) Quantum search on the two-dimensional lattice using the staggered model with hamiltonians. Phys Rev A 95:042341
Portugal R, Santos RAM, Fernandes TD, Gonçalves DN (2015) The staggered quantum walk model. Quantum Inf Process 15(1):85–101
Portugal R, de Oliveira MC, Moqadam JK (2017) Staggered quantum walks with hamiltonians. Phys Rev A 95:012328
Szegedy M (2004) Quantum speed-up of Markov chain based algorithms. In: Proceedings of the 45th symposium on foundations of computer science, pp 32–41
Acknowledgements
The author thanks Renato Portugal for useful discussions and comments, Bruno Chagas for sharing some calculations on SQWs and the anonymous reviewers for interesting suggestions to improve the manuscript. This work was supported by ERDF Project Number 1.1.1.2/VIAA/1/16/002.
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Santos, R.A.M. The role of tessellation intersection in staggered quantum walks. Nat Comput 19, 843–852 (2020). https://doi.org/10.1007/s11047-019-09758-2
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DOI: https://doi.org/10.1007/s11047-019-09758-2