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Walking on vertices and edges by continuous-time quantum walk

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Abstract

The quantum walk dynamics obey the laws of quantum mechanics with an extra locality constraint, which demands that the evolution operator is local in the sense that the walker must visit the neighboring locations before endeavoring to distant places. Usually, the Hamiltonian is obtained from either the adjacency or the Laplacian matrix of the graph and the walker hops from vertices to neighboring vertices. In this work, we define a version of the continuous-time quantum walk that allows the walker to hop from vertices to edges and vice versa. As an application, we analyze the spatial search algorithm on the complete bipartite graph by modifying the new version of the Hamiltonian with an extra term that depends on the location of the marked vertex or marked edge, similar to what is done in the standard continuous-time quantum walk model. We show that the optimal running time to find either a vertex or an edge is \(O(\sqrt{N_\textrm{e}})\) with success probability \(1-o(1)\), where \(N_\textrm{e}\) is the number of edges of the complete bipartite graph.

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References

  1. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)

    Article  ADS  Google Scholar 

  2. Portugal, R.: Quantum Walks and Search Algorithms, 2nd edn. Springer, Cham (2018)

    Book  MATH  Google Scholar 

  3. Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of the 33th STOC, pp. 50–59. ACM, New York (2001)

  4. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  5. Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quantum Inf. Process. 15(1), 85–101 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Higuchi, Y., Konno, N., Sato, I., Segawa, E.: Spectral and asymptotic properties of Grover walks on crystal lattices. J. Funct. Anal. 267(11), 4197–4235 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–41 (2004)

  8. Matsue, K., Ogurisu, O., Segawa, E.: Quantum walks on simplicial complexes. Quantum Inf. Process. 15(5), 1865–1896 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Zhan, H.: Quantum walks on embeddings. J. Algebr. Comb. 53(4), 1187–1213 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)

    Article  ADS  Google Scholar 

  11. Shenvi, N., Kempe, J., Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)

    Article  ADS  Google Scholar 

  12. Delvecchio, M., Groiseau, C., Petiziol, F., Summy, G.S., Wimberger, S.: Quantum search with a continuous-time quantum walk in momentum space. J. Phys. B At. Mol. Opt. Phys. 53(6), 065301 (2020)

    Article  ADS  Google Scholar 

  13. Wang, K., Shi, Y., Xiao, L., Wang, J., Joglekar, Y.N., Xue, P.: Experimental realization of continuous-time quantum walks on directed graphs and their application in PageRank. Optica 7(11), 1524–1530 (2020)

    Article  ADS  Google Scholar 

  14. Benedetti, C., Tamascelli, D., Paris, M.G.A., Crespi, A.: Quantum spatial search in two-dimensional waveguide arrays. Phys. Rev. Appl. 16, 054036 (2021)

    Article  ADS  Google Scholar 

  15. Qu, D., Marsh, S., Wang, K., Xiao, L., Wang, J., Xue, P.: Deterministic search on star graphs via quantum walks. Phys. Rev. Lett. 128, 050501 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  16. Gao, X., Luo, Y., Liu, W.: Kirchhoff index in line, subdivision and total graphs of a regular graph. Discrete Appl. Math. 160(4), 560–565 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, Y., Gu, R., Lei, H.: The generalized connectivity of the line graph and the total graph for the complete bipartite graph. Appl. Math. Comput. 347, 645–652 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Zhao, Shu-Li., Hao, Rong-Xia., Wei, C.: Internally disjoint trees in the line graph and total graph of the complete bipartite graph. Appl. Math. Comput. 422, 126990 (2022)

    MathSciNet  MATH  Google Scholar 

  19. Dündar, P., Aytaç, A.: Integrity of total graphs via certain parameters. Math. Notes 76(5), 665–672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cvetković, D.M.: Spectrum of the total graph of a graph. Publ. l’Inst. Math. 16(30), 49–52 (1973)

    MathSciNet  MATH  Google Scholar 

  21. Liu, X., Wang, Q.: Laplacian state transfer in total graphs. Discrete Math. 344(1), 112139 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hazama, F.: On the kernels of the incidence matrices of graphs. Discrete Math. 254, 165–174 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Akbari, S., Ghareghani, N., Khosrovshahi, G., Maimani, H.: The kernels of the incidence matrices of graphs revisited. Linear Algebra Appl. 414, 617–625 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Abreu, A., Cunha, L., de Figueiredo, C., Marquezino, F., Posner, D., Portugal, R.: Total tessellation cover and quantum walk. arXiv:2002.08992 (2020)

  25. Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. London Mathematical Society Student Texts, Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  26. Cvetković, D.: Spectra of graphs formed by some unary operations. Publ. l’Inst. Math. 19(33), 37–41 (1975)

    MathSciNet  MATH  Google Scholar 

  27. Lugão, P.H.G., Portugal, R., Sabri, M., Tanaka, H.: Multimarked spatial search by continuous-time quantum walk. arXiv:2203.14384 (2022)

  28. Bezerra, G.A., Lugão, P.H.G., Portugal, R.: Quantum-walk-based search algorithms with multiple marked vertices. Phys. Rev. A 103, 062202 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  29. Chan, A., Godsil, C., Tamon, C., Xie, W.: Of shadows and gaps in spatial search. arXiv:2204.04355 (2022)

  30. Tanaka, H., Sabri, M., Portugal, R.: Spatial search on Johnson graphs by continuous-time quantum walk. Quantum Inf. Process. 21(2), 74 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett. 112, 210502 (2014)

    Article  ADS  Google Scholar 

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Acknowledgements

The work of C. F. T. da Silva was supported by CNPq Grant Number 141985/2019-4. The work of R. Portugal was supported by FAPERJ Grant Number CNE E-26/202.872/2018, and CNPq Grant Number 308923/2019-7. The authors have no competing interests to declare that are relevant to the content of this article.

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Authors and Affiliations

  1. National Laboratory of Scientific Computing (LNCC), Petrópolis, RJ, 25651-075, Brazil

    Cauê F. Teixeira da Silva & Renato Portugal

  2. Universidade Federal Rural do Rio de Janeiro (UFRRJ), Seropédica, RJ, 23.897-000, Brazil

    Daniel Posner

  3. Senai Cimatec, Salvador, BA, 41650-010, Brazil

    Renato Portugal

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  1. Cauê F. Teixeira da Silva
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  2. Daniel Posner
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  3. Renato Portugal
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Correspondence to Renato Portugal.

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Silva, C.F.T.d., Posner, D. & Portugal, R. Walking on vertices and edges by continuous-time quantum walk. Quantum Inf Process 22, 93 (2023). https://doi.org/10.1007/s11128-023-03842-4

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