Document Open Access Logo

Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer

Authors Stacey Jeffery , Galina Pass

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.STACS.2025.54.pdf
  • Filesize: 0.81 MB
  • 16 pages
HTML5 Logo HTML (experimental)
LIPIcs.STACS.2025.54.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Quantum Divide & Conquer
  • Time-Efficient
  • Subspace Graphs
  • Quantum Walks
  • Switching Networks
  • Directed st-Connectivity

Metrics

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

Abstract

We introduce an object called a subspace graph that formalizes the technique of multidimensional quantum walks. Composing subspace graphs allows one to seamlessly combine quantum and classical reasoning, keeping a classical structure in mind, while abstracting quantum parts into subgraphs with simple boundaries as needed. As an example, we show how to combine a switching network with arbitrary quantum subroutines, to compute a composed function. As another application, we give a time-efficient implementation of quantum Divide & Conquer when the sub-problems are combined via a Boolean formula. We use this to quadratically speed up Savitch’s algorithm for directed st-connectivity.

Cite As Get BibTex

Stacey Jeffery and Galina Pass. Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.54

Author Details

Stacey Jeffery
  • QuSoft, CWI, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands
Galina Pass
  • QuSoft, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands

Funding

  • Jeffery, Stacey: Supported by ERC STG grant 101040624-ASC-Q, NWO Klein project number OCENW.Klein.061. SJ is a CIFAR Fellow in the Quantum Information Science Program.
  • Pass, Galina: Supported by the National Agenda for Quantum Technologies (NAQT), as part of the Quantum Delta NL programme.

References

  1. Jonathan Allcock, Jinge Bao, Aleksandrs Belovs, Troy Lee, and Miklos Santha. On the quantum time complexity of divide and conquer. https://arxiv.org/abs/2311.16401, 2023.
  2. Andris Ambainis. Quantum search with variable times. Theory of Computing Systems, 47:786-807, 2010. https://arxiv.org/abs/quant-ph/0609168. URL: https://doi.org/10.1007/s00224-009-9219-1.
  3. Simon Apers, Stacey Jeffery, Galina Pass, and Michael Walter. (No) Quantum Space-Time Tradeoff for USTCON. In 31st Annual European Symposium on Algorithms (ESA 2023), pages 10:1-10:17, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.10.
  4. Aleksandrs Belovs. Quantum walks and electric networks. https://arxiv.org/abs/1302.3143, 2013.
  5. Aleksandrs Belovs, Andrew M. Childs, Stacey Jeffery, Robin Kothari, and Frédéric Magniez. Time-efficient quantum walks for 3-distinctness. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP), pages 105-122, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_10.
  6. Aleksandrs Belovs, Stacey Jeffery, and Duyal Yolcu. Taming quantum time complexity. https://arxiv.org/abs/2311.15873, 2023. URL: https://doi.org/10.48550/arXiv.2311.15873.
  7. Aleksandrs Blovs and Ben W. Reichardt. Span programs and quantum algorithms for st-connectivity and claw detection. In Proceedings of the 20th Annual European Symposium on Algorithms (ESA), pages 193-204, 2012. URL: https://doi.org/10.1007/978-3-642-33090-2_18.
  8. M. L. Bonet and S. R. Buss. Size-depth tradeoffs for boolean formulae. Information Processing Letters, 49:151-155, 1994. URL: https://doi.org/10.1016/0020-0190(94)90093-0.
  9. Andrew M. Childs, Robin Kothari, Matt Kovacs-Deak, Aarthi Sundaram, and Daochen Wang. Quantum divide and conquer. https://arxiv.org/abs/2210.06419, 2022.
  10. Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. SIAM Journal on Computing, 35(6):1310-1328, 2006. Earlier version in ICALP'04. https://arxiv.org/abs/quant-ph/0401091. URL: https://doi.org/10.1137/050644719.
  11. Tsuyoshi Ito and Stacey Jeffery. Approximate span programs. Algorithmica, 79:2158-2195, 2019. URL: https://doi.org/10.1007/S00453-018-0527-1.
  12. Michael Jarret, Stacey Jeffery, Shelby Kimmel, and Alvaro Piedrafita. Quantum algorithms for connectivity and related problems. In Proceedings of the 26th Annual European Symposium on Algorithms (ESA), pages 49:1-49:13, 2018. URL: https://doi.org/10.4230/LIPIcs.ESA.2018.49.
  13. Stacey Jeffery. Quantum subroutine composition. https://arxiv.org/abs/2209.14146, 2022.
  14. Stacey Jeffery and Shelby Kimmel. Quantum algorithms for graph connectivity and formula evaluation. Quantum, 1(26), 2017. Google Scholar
  15. Stacey Jeffery and Sebastian Zur. Multidimensional quantum walks and application to k-distinctness. In Proceedings of the 55th ACM Symposium on the Theory of Computing (STOC), pages 1125-1130, 2023. URL: https://arxiv.org/abs/2208.13492.
  16. C. Y. Lee. Representation of switching functions by binary decision programs. Bell Systems Technical Journal, 38(4):985-999, 1959. URL: https://doi.org/10.1002/j.1538-7305.1959.tb01585.x.
  17. Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha. Search via quantum walk. SIAM Journal on Computing, 40(1):142-164, 2011. Earlier version in STOC'07. https://arxiv.org/abs/quant-ph/0608026. URL: https://doi.org/10.1137/090745854.
  18. Aaron H. Potechin. Analyzing monotone space complexity via the switching network model. PhD thesis, Massachusetts Institute of Technology, 2015. Google Scholar
  19. Ben W. Reichardt. Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), pages 544-551, 2009. https://arxiv.org/abs/0904.2759. URL: https://doi.org/10.1109/FOCS.2009.55.
  20. Ben W. Reichardt. Faster quantum algorithm for evaluating game trees. In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 546-559, 2011. URL: https://doi.org/10.5555/2133036.2133079.
  21. Ben W. Reichardt and Robert Špalek. Span-program-based quantum algorithm for evaluating formulas. Theory of Computing, 8(13):291-319, 2012. URL: https://doi.org/10.4086/toc.2012.v008a013.
  22. Walter J Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of computer and system sciences, 4(2):177-192, 1970. URL: https://doi.org/10.1016/S0022-0000(70)80006-X.
  23. Claude E. Shannon. A symbolic analysis of relay and switching networks. Transactions of the American Institute of Electrical Engineers, 57(12):713-723, 1938. URL: https://doi.org/10.1109/T-AIEE.1938.5057767.
  24. Claude E. Shannon. The synthesis of two-terminal switching circuits. Bell System Technical Journal, 28(1):59-98, 1949. URL: https://doi.org/10.1002/j.1538-7305.1949.tb03624.x.
  25. Mario Szegedy. Quantum speed-up of Markov chain based algorithms. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS), pages 32-41, 2004. https://arxiv.org/abs/quant-ph/0401053. URL: https://doi.org/10.1109/FOCS.2004.53.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact