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Szegedy’s quantum walk with queries

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Abstract

When searching for a marked vertex in a graph, Szegedy’s usual search operator is defined by using the transition probability matrix of the random walk with absorbing barriers at the marked vertices. Instead of using this operator, we analyze searching with Szegedy’s quantum walk by using reflections around the marked vertices, that is, the standard form of quantum query. We show we can boost the probability to 1 of finding a marked vertex in the complete graph. Numerical simulations suggest that the success probability can be improved for other graphs, like the two-dimensional grid. We also prove that, for a certain class of graphs, we can express Szegedy’s search operator, obtained from the absorbing walk, using the standard query model.

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Notes

  1. A strongly regular graph is a regular graph where every two adjacent vertices have \(\lambda \) common neighbors and every two non-adjacent vertices have \(\mu \) common neighbors.

References

  1. Shenvi, N., Kempe, J., Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67, 1–11 (2003)

  2. Childs, A., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70, 1–11 (2004)

  3. Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (2004)

  4. Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 1099–1108 (2005)

  5. Krovi, H., Magniez, F., Ozols, M., Roland, J.: Finding is as easy as detecting for quantum walks. In: Proceedings of the 37th International Colloquium Conference on Automata, Languages and Programming, pp. 540–551 (2010)

  6. Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)

    Book  MATH  Google Scholar 

  7. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687–1690 (1993)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

  10. Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. SIAM J. Comput. 40(1), 142–164 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Santos, R.A.M., Portugal, R.: Quantum hitting time on the complete graph. Int. J. Quant. Inf. 8(5), 881–894 (2010)

    Article  MATH  Google Scholar 

  12. Portugal, R., Santos, R.A.M., Fernandes, T.D., Gonçalves, D.N.: The staggered quantum walk model. Quant. Inf. Process. 15(1), 85–101 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Portugal, R.: Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model. Quant. Inf. Process. 5(4), 1387–1409 (2016)

  14. Wong, T.G.: Grover search with lackadaisical quantum walks. J. Phys. A. Mathe. Theor. 48(43), 435304 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Falk, M.: Quantum search on the spatial grid. arXiv:1303.4127 [quant-ph] (2013)

  16. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th ACM Symposium on the Theory of Computing, pp. 212–219 (1996)

  17. Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26(5), 1510–1523 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Prūsis, K., Vihrovs, J., Wong, T.G.: Doubling the success of quantum walk search using internal-state measurements. arXiv:1511.03865 [quant-ph] (2015)

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Acknowledgments

The author thanks R. Portugal and A. Ambainis for useful comments.

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Authors and Affiliations

  1. Faculty of Computing, University of Latvia, Raina bulv. 19, Riga, LV-1586, Latvia

    Raqueline A. M. Santos

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  1. Raqueline A. M. Santos
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Correspondence to Raqueline A. M. Santos.

Additional information

This work was supported by the European Union Seventh Framework Programme (FP7/2007-2013) under the QALGO (Grant Agreement No. 600700) project.

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Santos, R.A.M. Szegedy’s quantum walk with queries. Quantum Inf Process 15, 4461–4475 (2016). https://doi.org/10.1007/s11128-016-1427-4

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  • DOI: https://doi.org/10.1007/s11128-016-1427-4

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