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Quantum Speedup for Sampling Random Spanning Trees

Authors Simon Apers, Minbo Gao, Zhengfeng Ji, Chenghua Liu

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Quantum computation theory
Keywords
  • Quantum Computing
  • Quantum Algorithms
  • Random Spanning Trees

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Abstract

We present a quantum algorithm for sampling random spanning trees from a weighted graph in Õ(√{mn}) time, where n and m denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in Õ(m) time. The approach carefully combines, on one hand, a classical method based on "large-step" random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi’s multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.

Cite As Get BibTex

Simon Apers, Minbo Gao, Zhengfeng Ji, and Chenghua Liu. Quantum Speedup for Sampling Random Spanning Trees. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.13

Author Details

Simon Apers
  • Université Paris Cité, CNRS, IRIF, Paris, France
Minbo Gao
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China
Zhengfeng Ji
  • Department of Computer Science and Technology, Tsinghua University, Beijing, China
Chenghua Liu
  • Key Laboratory of System Software (Chinese Academy of Sciences) and State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China
  • University of Chinese Academy of Sciences, Beijing, China

Funding

The work is supported by National Key Research and Development Program of China (Grant No. 2023YFA1009403), National Natural Science Foundation of China (Grant No. 12347104), and Beijing Natural Science Foundation (Grant No. Z220002). SA was supported in part by the European QuantERA project QOPT (ERA-NET Cofund 2022-25), the French PEPR integrated projects EPiQ (ANR-22-PETQ-0007) and HQI (ANR-22-PNCQ-0002), and the French ANR project QUOPS (ANR-22-CE47-0003-01).

Acknowledgements

We would like to thank Yang P. Liu for explaining the details of their work. Minbo Gao and Chenghua Liu would like to thank Sebastian Zur for helpful discussions on quantum random walks and quantum sampling algorithms. Minbo Gao would like to thank Jingqin Yang for discussions of link-cut trees used in previous spanning tree sampling algorithms.

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