Document Open Access Logo

Improved Algorithm and Lower Bound for Variable Time Quantum Search

Authors Andris Ambainis , Martins Kokainis , Jevgēnijs Vihrovs

PDF
Thumbnail PDF

File

LIPIcs.TQC.2023.7.pdf
  • Filesize: 0.78 MB
  • 18 pages

Document Identifiers

Author Details

Andris Ambainis
  • Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Martins Kokainis
  • Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Jevgēnijs Vihrovs
  • Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia

Funding

This research was supported by the ERDF project 1.1.1.5/18/A/020.

Acknowledgements

We thank Krišjānis Prūsis for useful discussions on the lower bound proof. The authors are grateful to the anonymous referees for the helpful comments and suggestions.

Cite AsGet BibTex

Andris Ambainis, Martins Kokainis, and Jevgēnijs Vihrovs. Improved Algorithm and Lower Bound for Variable Time Quantum Search. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.TQC.2023.7

Abstract

We study variable time search, a form of quantum search where queries to different items take different time. Our first result is a new quantum algorithm that performs variable time search with complexity O(√Tlog n) where T = ∑_{i = 1}ⁿ t_i² with t_i denoting the time to check the i^th item. Our second result is a quantum lower bound of Ω(√{Tlog T}). Both the algorithm and the lower bound improve over previously known results by a factor of √{log T} but the algorithm is also substantially simpler than the previously known quantum algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum search
  • amplitude amplification

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Scott Aaronson. Lower bounds for local search by quantum arguments. SIAM Journal on Computing, 35(4):804-824, 2006. https://arxiv.org/abs/quant-ph/0307149. URL: https://doi.org/10.1137/S0097539704447237.
  2. Andris Ambainis. Quantum search with variable times. Theory of Computing Systems, 47(3):786-807, 2010. https://arxiv.org/abs/quant-ph/0609168. URL: https://doi.org/10.1007/s00224-009-9219-1.
  3. Andris Ambainis. Variable time amplitude amplification and quantum algorithms for linear algebra problems. In Christoph Dürr and Thomas Wilke, editors, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012), volume 14 of Leibniz International Proceedings in Informatics (LIPIcs), pages 636-647. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. https://arxiv.org/abs/1010.4458. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.636.
  4. Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail. Provably secure key establishment against quantum adversaries. In Mark M. Wilde, editor, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), volume 73 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1-3:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. https://arxiv.org/abs/1704.08182. URL: https://doi.org/10.4230/LIPIcs.TQC.2017.3.
  5. Gilles Brassard. Searching a quantum phone book. Science, 275(5300):627-628, 1997. URL: https://doi.org/10.1126/science.275.5300.627.
  6. Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53-74, 2002. https://arxiv.org/abs/quant-ph/0005055. URL: https://doi.org/10.1090/conm/305/05215.
  7. Sourav Chakraborty, Arkadev Chattopadhyay, Peter Høyer, Nikhil S. Mande, Manaswi Paraashar, and Ronald de Wolf. Symmetry and quantum query-to-communication simulation. In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), volume 219 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:23, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://arxiv.org/abs/2012.05233. URL: https://doi.org/10.4230/LIPIcs.STACS.2022.20.
  8. Andrew M. Childs, Robin Kothari, and Rolando D. Somma. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM Journal on Computing, 46(6):1920-1950, 2017. https://arxiv.org/abs/1511.02306. URL: https://doi.org/10.1137/16M1087072.
  9. Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita. Span programs and quantum time complexity. In Javier Esparza and Daniel Kráľ, editors, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020), volume 170 of Leibniz International Proceedings in Informatics (LIPIcs), pages 26:1-26:14, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. https://arxiv.org/abs/2005.01323. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.26.
  10. Koen de Boer, Léo Ducas, Stacey Jeffery, and Ronald de Wolf. Attacks on the AJPS Mersenne-based cryptosystem. In Tanja Lange and Rainer Steinwandt, editors, Post-Quantum Cryptography, pages 101-120, Cham, 2018. Springer International Publishing. Preprint: https://eprint.iacr.org/2017/1171. URL: https://doi.org/10.1007/978-3-319-79063-3_5.
  11. Adam Glos, Martins Kokainis, Ryuhei Mori, and Jevgēnijs Vihrovs. Quantum speedups for dynamic programming on n-dimensional lattice graphs. In Filippo Bonchi and Simon J. Puglisi, editors, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), volume 202 of Leibniz International Proceedings in Informatics (LIPIcs), pages 50:1-50:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. https://arxiv.org/abs/2104.14384. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.50.
  12. Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103:150502, 2009. https://arxiv.org/abs/0811.3171. URL: https://doi.org/10.1103/PhysRevLett.103.150502.
  13. Peter Høyer, Michele Mosca, and Ronald de Wolf. Quantum search on bounded-error inputs. In Jos C. M. Baeten, Jan Karel Lenstra, Joachim Parrow, and Gerhard J. Woeginger, editors, Automata, Languages and Programming, pages 291-299, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. https://arxiv.org/abs/quant-ph/0304052. URL: https://doi.org/10.1007/3-540-45061-0_25.
  14. Stacey Jeffery. Quantum subroutine composition, 2022. URL: https://arxiv.org/abs/2209.14146.
  15. François Le Gall. Improved quantum algorithm for triangle finding via combinatorial arguments. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 216-225. IEEE, 2014. https://arxiv.org/abs/1407.0085. URL: https://doi.org/10.1109/FOCS.2014.31.
  16. André Schrottenloher and Marc Stevens. A quantum analysis of nested search problems with applications in cryptanalysis. Cryptology ePrint Archive, Paper 2022/761, 2022. URL: https://eprint.iacr.org/2022/761.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact