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Directed st-Connectivity with Few Paths Is in Quantum Logspace

Authors Simon Apers , Roman Edenhofer

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum computation
  • Space-bounded complexity classes
  • Graph connectivity
  • Unambiguous computation
  • Random walks

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Abstract

We present a BQSPACE(O(log n))-procedure to count st-paths on directed graphs for which we are promised that there are at most polynomially many paths starting in s and polynomially many paths ending in t. For comparison, the best known classical upper bound in this case just to decide st-connectivity is DSPACE(O(log² n/ log log n)). The result establishes a new relationship between BQL and unambiguity and fewness subclasses of NL. Further, we also show how to recognize directed graphs with at most polynomially many paths between any two nodes in BQSPACE(O(log n)). This yields the first natural candidate for a language separating BQL from 𝖫 and BPL. Until now, all candidates potentially separating these classes were inherently promise problems.

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Simon Apers and Roman Edenhofer. Directed st-Connectivity with Few Paths Is in Quantum Logspace. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.18

Author Details

Simon Apers
  • Université Paris Cité, CNRS, IRIF, Paris, France
Roman Edenhofer
  • Université Paris Cité, CNRS, IRIF, Paris, France

Funding

  • Apers, Simon: EPIQ (ANR-22-PETQ-0007), QUDATA (ANR18-CE47-0010), QUOPS (ANR-22-CE47-0003-01) and HQI (ANR-22-PNCQ-0002), and EU project QOPT (QuantERA ERA-NET Cofund 2022-25).
  • Edenhofer, Roman: This work has received support under the program "Investissement d'Avenir" launched by the French Government and implemented by ANR, with the reference "ANR‐22‐CMAS-0001, QuanTEdu-France".

Acknowledgements

We thank François Le Gall for discussions about the differences between language and promise classes. He made us aware that our results can be seen as first evidence that the language classes BQL and BPL are distinct. We further acknowledge useful discussions with Klaus-Jörn Lange and we thank Lior Eldar, Troy Lee and Ronald de Wolf for valuable comments on an earlier draft.

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