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Quantum Walks on Two-Dimensional Grids with Multiple Marked Locations

  • Nikolajs NahimovsEmail author
  • Alexander Rivosh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9587)

Abstract

The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for an arbitrary configuration of marked locations.

In this paper, we prove two results showing the importance of how the marked locations are arranged. First, we present two placements of k marked locations for which the number of steps of the algorithm differs by a factor of \(\varOmega (\sqrt{k})\). Second, we present two configurations of k and \(\sqrt{k}\) marked locations having the same number of steps and probability to find a marked location.

References

  1. 1.
    Ambainis, A., Bačkurs, A., Nahimovs, N., Ozols, R., Rivosh, A.: Search by quantum walks on two-dimensional grid without amplitude amplification. In: Kawano, Y. (ed.) TQC 2012. LNCS, vol. 7582, pp. 87–97. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37, 210–239 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambainis, A., Kempe, J., Rivosh, A.: Coins make quantum walks faster. In: Proceedings of SODA 2005, pp. 1099–1108 (2005)Google Scholar
  4. 4.
    Ambainis, A., Portugal, R., Nahimov, N.: Spatial search on grids with minimum memory (2014). arXiv:1312.0172
  5. 5.
    Buhrman, H., Spalek, R.: Quantum verification of matrix products. In: Proceedings SODA 2006, pp. 880–889 (2006)Google Scholar
  6. 6.
    Krovi, H., Magniez, F., Ozols, M., Roland, J.: Finding is as easy as detecting for quantum walks. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 540–551. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Krov, H.: Symmetry in quantum walks (Ph.D thesis) (2007). arXiv:0711.1694
  8. 8.
    Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett. 112, 210502 (2014). arXiv:1403.2228 CrossRefGoogle Scholar
  9. 9.
    Meyer, D.A., Wong, T.G.: Connectivity is a poor indicator of fast quantum search. Phys. Rev. Lett. 114, 110503 (2015). arXiv:1409.5876 CrossRefGoogle Scholar
  10. 10.
    Magniez, F., Santha, M., Szegedy, M.: An \(O(n^{1.3})\) quantum algorithm for the triangle problem. In: Proceedings of SODA 2005, pp. 413–424 (2005)Google Scholar
  11. 11.
    Portugal, R.: Quantum Walks and Search Algorithms. Springer, New York (2013)zbMATHCrossRefGoogle Scholar
  12. 12.
    Shenvi, N., Kempe, J., Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5), 052307 (2003)CrossRefGoogle Scholar
  13. 13.
    Szegedy, M: Quantum speed-up of markov chain based algorithms. In: Proceedings of FOCS 2004, pp. 32–41 (2004)Google Scholar
  14. 14.
    Wong, T.G.: On the Breakdown of Quantum Search with Spatially Distributed Marked Vertices vol. 1501, p. 07071 (2015)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Faculty of ComputingUniversity of LatviaRigaLatvia

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