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Adjacent Vertices Can Be Hard to Find by Quantum Walks

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SOFSEM 2017: Theory and Practice of Computer Science (SOFSEM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10139))

Abstract

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs \(\varOmega (N)\) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph.

This work was supported by the European Union Seventh Framework Programme (FP7/2007-2013) under the QALGO (Grant Agreement No. 600700) project and the RAQUEL (Grant Agreement No. 323970) project, the Latvian State Research Programme NeXIT project No. 1 and the ERC Advanced Grant MQC.

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Acknowledgements

The authors thank Andris Ambainis for helpful ideas and suggestions, which was a great help during this research, and Tom Wong for useful comments and careful revision of the manuscript.

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Authors and Affiliations

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

    Nikolajs Nahimovs & Raqueline A. M. Santos

Authors
  1. Nikolajs Nahimovs
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  2. Raqueline A. M. Santos
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Corresponding author

Correspondence to Raqueline A. M. Santos .

Editor information

Editors and Affiliations

  1. TU Dortmund , Dortmund, Germany

    Bernhard Steffen

  2. TU Dresden , Dresden, Germany

    Christel Baier

  3. Eindhoven University of Technology , Eindhoven, The Netherlands

    Mark van den Brand

  4. Alpen Adria University Klagenfurt , Klagenfurt, Austria

    Johann Eder

  5. Lero - Irish Software Research Center , Limerick, Ireland

    Mike Hinchey

  6. Lero - Irish Software Research Center , Limerick, Ireland

    Tiziana Margaria

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Nahimovs, N., Santos, R.A.M. (2017). Adjacent Vertices Can Be Hard to Find by Quantum Walks. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds) SOFSEM 2017: Theory and Practice of Computer Science. SOFSEM 2017. Lecture Notes in Computer Science(), vol 10139. Springer, Cham. https://doi.org/10.1007/978-3-319-51963-0_20

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  • DOI: https://doi.org/10.1007/978-3-319-51963-0_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-51962-3

  • Online ISBN: 978-3-319-51963-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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