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(No) Quantum Space-Time Tradeoff for USTCON

Authors Simon Apers, Stacey Jeffery, Galina Pass, Michael Walter

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Author Details

Simon Apers
  • CNRS, IRIF, Paris, France
Stacey Jeffery
  • CWI & QuSoft, Amsterdam, The Netherlands
Galina Pass
  • Korteweg-de Vries Institute for Mathematics & QuSoft, University of Amsterdam, The Netherlands
  • Faculty of Computer Science, Ruhr University Bochum, Germany
Michael Walter
  • Faculty of Computer Science, Ruhr University Bochum, Germany

Funding

  • Jeffery, Stacey: Supported by ERC STG grant 101040624-ASC-Q, NWO Klein project number OCENW.Klein.061, and ARO contract no W911NF2010327. 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.
  • Walter, Michael: Supported by the European Research Council (ERC) through ERC Starting Grant 101040907-SYMOPTIC, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972, the Federal Ministry of Education and Research (BMBF) through project Quantum Methods and Benchmarks for Resource Allocation (QuBRA), and NWO grant OCENW.KLEIN.267.

Acknowledgements

Part of this work was initiated at the 2022 QOPT (QuantERA ERA-NET Cofund 2022-25) workshop hosted at Université Libre de Bruxelles.

Cite AsGet BibTex

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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.10

Abstract

Undirected st-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of T = Õ(n²/S) for any S such that S = Ω(log(n)) and S = O(n²/m). Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time Õ(n) and space O(log(n)) simultaneously. This improves on previous results, which required either O(log(n)) space and Õ(n^{1.5}) time, or Õ(n) space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Undirected st-connectivity
  • quantum walks
  • time-space tradeoff

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References

  1. Dorit Aharonov and Amnon Ta-Shma. Adiabatic quantum state generation and statistical zero knowledge. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 20-29, 2003. Google Scholar
  2. Romas Aleliunas, Richard M. Karp, Richard J. Lipton, Laszlo Lovasz, and Charles Rackoff. Random walks, universal traversal sequences, and the complexity of maze problems. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science (FOCS), pages 218-223, 1979. URL: https://doi.org/10.1109/SFCS.1979.34.
  3. Simon Apers. Quantum walk sampling by growing seed sets. In Proceedings of the 27th Annual European Symposium on Algorithms (ESA), volume 144, pages 9:1-9:12. Springer, 2019. Google Scholar
  4. Simon Apers, András Gilyén, and Stacey Jeffery. A unified framework of quantum walk search. In Proceedings of the 38th Symposium on Theoretical Aspects of Computer Science (STACS), pages 6:1-6:13, 2020. https://arxiv.org/abs/1912.04233. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.6.
  5. Greg Barnes and Uriel Feige. Short random walks on graphs. In Proceedings of the 1993 ACM Symposium on the Theory of Computing (STOC), pages 728-737, 1993. Google Scholar
  6. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. Earlier version in FOCS'98. https://arxiv.org/abs/quant-ph/9802049. URL: https://doi.org/10.1145/502090.502097.
  7. Paul Beame, Allan Borodin, Prabhakar Raghavan, Walter L Ruzzo, and Martin Tompa. Time-space tradeoffs for undirected graph traversal. In Proceedings of the 1990 IEEE Symposium on Foundations of Computer Science (FOCS), pages 429-438. IEEE, 1990. Google Scholar
  8. Aleksandrs Belovs. Quantum walks and electric networks. https://arxiv.org/abs/1302.3143, 2013.
  9. Aleksandrs Belovs 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.
  10. Graham Brightwell and Peter Winkler. Maximum hitting time for random walks on graphs. Random Structures & Algorithms, 1(3):263-276, 1990. Google Scholar
  11. Andrei Z Broder, Anna R Karlin, Prabhakar Raghavan, and Eli Upfal. Trading space for time in undirected st-connectivity. In Proceedings of the 1989 ACM Symposium on the Theory of Computing (STOC), pages 543-549, 1989. Google Scholar
  12. Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca. Quantum algorithms revisited. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1969):339-354, 1998. Google Scholar
  13. 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.
  14. Jeff Edmonds. Time-space trade-offs for undirected st-connectivity on a JAG. In Proceedings of the 1993 ACM Symposium on the Theory of Computing (STOC), pages 718-727, 1993. Google Scholar
  15. Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Limit on the speed of quantum computation in determining parity. Physical Review Letters, 81(24):5442, 1998. https://arxiv.org/abs/quant-ph/9802045. URL: https://doi.org/10.1103/PhysRevLett.81.5442.
  16. Uriel Feige. A randomized time-space tradeoff of Õ(m R̂) for USTCON. In Proceedings of the 1993 IEEE Symposium on Foundations of Computer Science (FOCS), pages 238-246. IEEE, 1993. Google Scholar
  17. Lov Grover and Terry Rudolph. Creating superpositions that correspond to efficiently integrable probability distributions. arXiv preprint quant-ph/0208112, 2002. Google Scholar
  18. Iordanis Kerenidis and Anupam Prakash. Quantum recommendation systems. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS), pages 49:1-49:21. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  19. Adrian Kosowski. Faster walks in graphs: A Õ(n²) time-space trade-off for undirected s-t connectivity. In Proceedings of the 2013 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1873-1883, 2013. https://arxiv.org/abs/1204.1136v2. URL: https://doi.org/10.1137/1.9781611973105.133.
  20. Jessica Lemieux, Bettina Heim, David Poulin, Krysta Svore, and Matthias Troyer. Efficient quantum walk circuits for metropolis-hastings algorithm. Quantum, 4:287, 2020. Google Scholar
  21. David A Levin, Yuval Peres, and Elizabeth L Wilmer. Markov chains and mixing times. American Mathematical Society, 2017. second edition. Google Scholar
  22. Omer Reingold. Undirected connectivity in log-space. Journal of the ACM, 55(4), 2008. URL: https://doi.org/10.1145/1391289.1391291.
  23. Avi Wigderson. The complexity of graph connectivity. In International Symposium on Mathematical Foundations of Computer Science, pages 112-132. Springer, 1992. Google Scholar
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