(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


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

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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)


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
  • Undirected st-connectivity
  • quantum walks
  • time-space tradeoff


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