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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 As Get 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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