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Quantum Search Algorithm for Exceptional Vertexes in Regular Graphs and its Circuit Implementation

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Abstract

There are some interesting properties in quantum mechanics, such as quantum superposition and entanglement. We can use these features to solve some specific problems. Quantum computer has more advantages than classical computer in many problems. In this paper, we are interested in regular graph. A special case of graph structure destroyed is given. We present a search algorithm to find exceptional vertexes in the graph in this case. The algorithm uses the trick of amplitude amplification in quantum search algorithm. In the corresponding classical algorithm, the adjacency matrix may be used to store the information of vertex and edge of graph. It takes one time for the best and N times for the worst to find the target. Which means that N/2 times on average need to be conducted. In our quantum algorithm, nodes and vertexes are stored in quantum states, using quantum search to design algorithm. The search process can be accelerated in our quantum algorithm.

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References

  1. Lov, K.: Grover: a fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing. STOC ’96, 212–219, New York (1996)

  2. Lov, K.: Grover: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328 (1997)

    Article  Google Scholar 

  3. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46(4-5), 493–505 (1998)

    Article  ADS  Google Scholar 

  4. Brassard, G., HØyer, P., Tapp, A.: Quantum Counting. In: Automata, Languages and Programming, pp 820–831. Springer, Berlin (1998)

  5. Peter, W.: Shor: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. J. Comput. 26(5), 1484–1509 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)

    Article  ADS  Google Scholar 

  7. Potočk, V., Gábris, A., Kiss, T., Jex, I.: Optimized quantum random-walk search algorithms on the hypercube. Phys. Rev. A 79, 012325 (2009)

    Article  ADS  Google Scholar 

  8. Reitzner, D., Hillery, M., Feldman, E., Bužek, V.: Quantum searches on highly symmetric graphs. Phys. Rev. A 79, 012323 (2009)

    Article  ADS  Google Scholar 

  9. Figgatt, C., Maslov, D., Landsman, K.A., Linke, N.M., Debnath, S., Monroe, C.: Complete 3-qubit grover search on a programmable quantum computer. Nat. Commun. 8, 1918 (2017)

    Article  ADS  Google Scholar 

  10. Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Daniel, A.: Spielman: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the thirty-fifth ACM symposium on Theory of computing STOC ’03. ACM Press (2003)

  11. Dörn, S.: Quantum algorithms for optimal graph traversal problems. Quantum Information and Computation V SPIE (2007)

  12. Djelloul, H., Layeb, A., Chikhi, S.: Quantum inspired cuckoo search algorithm for graph colouring problem. Int. J. Bio-Inspired Computat. 7(3), 183 (2015)

    Article  Google Scholar 

  13. Dürr, C., Heiligman, M., Høyer, P., Mhalla, M.: Quantum Query Complexity of Some Graph Problems. In: Automata, Languages and Programming, pp 481–493. Springer, Berlin (2004)

  14. Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM J. Comput. 37(2), 413–424 (2007)

    Article  MathSciNet  Google Scholar 

  15. Childs, A.M., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Inf. Process 1(1/2), 35–43 (2002)

    Article  MathSciNet  Google Scholar 

  16. Keating, J.P., Linden, N., Matthews, J.C.F.: A. Winter: Localization and its consequences for quantum walk algorithms and quantum communication. Phys. Rev. A 76(1), 012315 (2007)

    Article  ADS  Google Scholar 

  17. Minello, G., Rossi, L., Torsello, A.: Can a quantum walk tell which is which?a study of quantum walk-based graph similarity. Entropy 21(3), 32 (2019)

    Article  MathSciNet  Google Scholar 

  18. Konno, N., Mitsuhashi, H., Sato, I.: The discrete-time quaternionic quantum walk on a graph. Quantum Inf. Process 15(2), 651–673 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  19. Emms, D., Wilson, R.C., Hancock, E.R.: Graph matching using the interference of continuous-time quantum walks. Pattern Recogn. 42(5), 985–1002 (2009)

    Article  ADS  Google Scholar 

  20. Emms, D., Hancock, E.R., Wilson, R.C.: a correspondence measure for graph matching using the discrete quantum walk. In: Proceedings of the 6th IAPR-TC-15 International Conference on Graph-Based Representations in Pattern Recognition GbRPR’07, pp 81–91. Berlin, Heidelberg (2007)

  21. Higuchi, Y., Konno, N., Sato, I., Segawa, E.: Periodicity of the discrete-time quantum walk on a finite graph. Interdiscip. Inf. Sci. 23(1), 75–86 (2017)

    MathSciNet  Google Scholar 

  22. Lu, C., Sun, G., Zhang, Y.: Stationary distribution and extinction of a multi-stage HIV model with nonlinear stochastic perturbation. J. Appl. Math. Comput. 1–23 (2021)

  23. Lopez Acevedo, O., Gobron, T.: Quantum walks on cayley graphs. J. Phys. A Math. General 39(3), 585–599 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  24. Wong, T.G.: Faster quantum walk search on a weighted graph. Phys. Rev. A 92(3), 032320 (2015)

    Article  ADS  Google Scholar 

  25. Tamascelli, D., Zanetti, L.: A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems. J. Phys. A Math. Theoret. 47(32), 325302 (2014)

    Article  MathSciNet  Google Scholar 

  26. Bogdan, A.V., Fadin, V.S.: Quark regge trajectory in two loops from unitarity relations. Phys. At. Nucl. 68(9), 1599–1615 (2005)

    Article  Google Scholar 

  27. Gamble, J.K., Friesen, M., Zhou, D., Joynt, R., Coppersmith, S.N.: Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A 81(5), 052313 (2010)

    Article  ADS  Google Scholar 

  28. Rudinger, K, Gamble, J.K., Wellons, M., Bach, E., Friesen, M., Joynt, R., Coppersmith, S.N.: Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs. Phys. Rev. A 86(2), 022334 (2012)

    Article  ADS  Google Scholar 

  29. Deutsch, D.: Quantum theory, the church–turing principle and the universal quantum computer. Proc. R. Soc. London. A. Math. Phys. Sci. 400(1818), 97–117 (1985)

    MathSciNet  MATH  ADS  Google Scholar 

  30. Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science IEEE Comput. Soc. Press (1994)

  31. Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. R. Soc. London. Ser. A Math. Phys. Eng. Sci. 454(1969), 339–354 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  32. Griffiths, R.B., Niu, C.-S.: Semiclassical fourier transform for quantum computation. Phys. Rev. Lett. 76(17), 3228–3231 (1996)

    Article  ADS  Google Scholar 

  33. Isaac, L.: Chuang michael a nielsen: quantum computation and quantum information. Cambridge, England (2011)

    Google Scholar 

  34. Moscafi, M.: Quantum searching, counting and amplitude amplification by eigenvector analysis. In: MFCS’98 Workshop on Randomized Algorithms, pp 90–100 (1998)

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Acknowledgements

The National Natural Science Foundation of China (No.61772295, 61572270, and 61173056). The PHD foundation of Chongqing Normal University(No.19XLB003). The Science and Technology Research Program of Chongqing Municipal Education Commission(Grant No.KJZD-M202000501). Chongqing Technology Innovation and application development special general project(cstc2020jscx-lyjsAX0002).

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Authors and Affiliations

  1. College of Computer and Information Science, Chongqing Normal University, Chongqing, 401331, China

    Yumin Dong, Zhixin Liu & Jinlei Zhang

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  1. Yumin Dong
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  2. Zhixin Liu
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  3. Jinlei Zhang
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Contributions

Yumin Dong: Methodology, Conceptualization, Original Draft, Project administration and Funding acquisition. Zhixin Liu: Methodology, Conceptualization, Original Draft. Jinlei Zhang: Validation, Formal analysis, Review and Editing.

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Correspondence to Yumin Dong.

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Dong, Y., Liu, Z. & Zhang, J. Quantum Search Algorithm for Exceptional Vertexes in Regular Graphs and its Circuit Implementation. Int J Theor Phys 60, 2723–2732 (2021). https://doi.org/10.1007/s10773-021-04861-6

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  • DOI: https://doi.org/10.1007/s10773-021-04861-6

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