Document Open Access Logo

Quantum Complexity of Minimum Cut

Authors Simon Apers, Troy Lee

Thumbnail PDF


  • Filesize: 0.92 MB
  • 33 pages

Document Identifiers

Author Details

Simon Apers
  • CWI, Amsterdam, The Netherlands
  • Université libre de Bruxelles (ULB), Brussels, Belgium
Troy Lee
  • Centre for Quantum Software and Information, University of Technology Sydney, Australia


We would like to thank Ronald de Wolf for discussions which started this paper, and in particular a conversation which led to Theorem 35. We also thank Debmalya Panigrahi and Miklos Santha for helpful conversations on this topic.

Cite AsGet BibTex

Simon Apers and Troy Lee. Quantum Complexity of Minimum Cut. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 28:1-28:33, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most τ, we give a quantum algorithm to solve the minimum cut problem using Õ(n^{3/2}√{τ}) queries and time. Moreover, for every integer 1 ≤ τ ≤ n we give an example of a graph G with edge weights 1 and τ such that solving the minimum cut problem on G requires Ω(n^{3/2}√{τ}) queries to the adjacency matrix of G. These results contrast with the classical randomized case where Ω(n²) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is ̃ Θ(m). We show that the quantum query and time complexity are Õ(√{mnτ}) and Õ(√{mnτ} + n^{3/2}), respectively, where again the edge weights are between 1 and τ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n^{3/2}) for τ > 1 and Ω(τ n) for any 1 ≤ τ ≤ n. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger’s tree packing technique (STOC 1996).

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Quantum complexity theory
  • Quantum algorithms
  • quantum query complexity
  • minimum cut


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Andris Ambainis and Robert Špalek. Quantum algorithms for matching and network flows. In Proceedings of the Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 172-183. Springer, 2006. Google Scholar
  2. Simon Apers and Ronald de Wolf. Quantum speedup for graph sparsification, cut approximation and Laplacian solving. In Proceedings of the 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 637-648. IEEE, 2020. Google Scholar
  3. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-Ramanujan sparsifiers. SIAM J. Comput., 41(6):1704-1721, 2012. URL:
  4. Aleksandrs Belovs, Andrew M Childs, Stacey Jeffery, Robin Kothari, and Frédéric Magniez. Time-efficient quantum walks for 3-distinctness. In International Colloquium on Automata, Languages, and Programming, pages 105-122. Springer, 2013. Google Scholar
  5. András A. Benczúr and David R. Karger. Randomized approximation schemes for cuts and flows in capacitated graphs. SIAM J. Comput., 44(2):290-319, 2015. URL:
  6. Nalin Bhardwaj, Antonio M. Lovett, and Bryce Sandlund. A simple algorithm for minimum cuts in near-linear time. In Proceedings of the 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), volume 162 of LIPIcs, pages 12:1-12:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  7. Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Query complexity of global minimum cut. CoRR, abs/2007.09202, 2020. URL:
  8. Rodrigo A. Botafogo. Cluster analysis for hypertext systems. In Proceedings of the 16th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, page 116–125, New York, NY, USA, 1993. Association for Computing Machinery. URL:
  9. Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude amplification and estimation. Quantum computation and quantum information: A millennium volume, 305, 2002. Google Scholar
  10. Harry Buhrman, Richard Cleve, Ronald de Wolf, and Christof Zalka. Bounds for small-error and zero-error quantum algorithms. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 358-368. IEEE, 1999. Google Scholar
  11. Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla. Quantum query complexity of some graph problems. SIAM J. Comput., 35(6):1310-1328, 2006. URL:
  12. Christoph Dürr and Peter Høyer. A quantum algorithm for finding the minimum. CoRR, quant-ph/9607014, 1996. URL:
  13. Lester R. Ford and Delbert R. Fulkerson. Flows in Networks. Princeton University Press, 1962. URL:
  14. Wai Shing Fung, Ramesh Hariharan, Nicholas J. A. Harvey, and Debmalya Panigrahi. A general framework for graph sparsification. SIAM J. Comput., 48(4):1196-1223, 2019. URL:
  15. Harold N. Gabow. A matroid approach to finding edge connectivity and packing arborescences. J. Comput. Syst. Sci., 50(2):259-273, 1995. URL:
  16. Paweł Gawrychowski, Shay Mozes, and Oren Weimann. Minimum cut in O(m log² n) time. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP), volume 168 of LIPIcs, pages 57:1-57:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  17. Paweł Gawrychowski, Shay Mozes, and Oren Weimann. A note on a recent algorithm for minimum cut. In Symposium on Simplicity in Algorithms (SOSA), pages 74-79. SIAM, 2021. Google Scholar
  18. Ralph E. Gomory and Te C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. URL:
  19. Lov Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett., 78:325-328, 1997. Google Scholar
  20. Monika R. Henzinger and Valerie King. Randomized dynamic graph algorithms with polylogarithmic time per operation. In Proceedings of the 27th annual ACM symposium on Theory of computing (STOC), pages 519-527, 1995. Google Scholar
  21. Monika R. Henzinger, Satish Rao, and Di Wang. Local flow partitioning for faster edge connectivity. SIAM J. Comput., 49(1):1-36, 2020. URL:
  22. Monika R. Henzinger and David P. Williamson. On the number of small cuts in a graph. Inf. Process. Lett., 59(1):41-44, 1996. URL:
  23. Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), pages 526-535. ACM, 2007. URL:
  24. David R. Karger. Random sampling in cut, flow, and network design problems. Mathematics of Operations Research, 24(2):383-413, 1999. URL:
  25. David R. Karger. Minimum cuts in near-linear time. J. ACM, 47(1):46-76, 2000. URL:
  26. Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic edge connectivity in near-linear time. J. ACM, 66(1):4:1-4:50, 2019. URL:
  27. Jason Li. Deterministic mincut in almost-linear time. In Proceedings of the 53rd annual ACM symposium on Theory of computing (STOC), 2021. Google Scholar
  28. On-Hei S. Lo, Jens M. Schmidt, and Mikkel Thorup. Compact cactus representations of all non-trivial min-cuts. Discrete Applied Mathematics, 2020. URL:
  29. David W. Matula. A linear time 2+epsilon approximation algorithm for edge connectivity. In Vijaya Ramachandran, editor, Proceedings of the 4th Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms (SODA), pages 500-504. ACM/SIAM, 1993. URL:
  30. Sagnik Mukhopadhyay and Danupon Nanongkai. Weighted min-cut: sequential, cut-query, and streaming algorithms. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 496-509, 2020. Google Scholar
  31. Hiroshi Nagamochi and Toshihide Ibaraki. Computing edge-connectivity in multigraphs and capacitated graphs. SIAM J. Discret. Math., 5(1):54-66, 1992. URL:
  32. Ashwin Nayak and Felix Wu. The quantum query complexity of approximating the median and related statistics. In Proceedings of the 31st annual ACM symposium on Theory of computing (STOC), pages 384-393, 1999. Google Scholar
  33. Jean-Claude Picard and Maurice Queyranne. Selected applications of minimum cuts in networks. INFOR: Information Systems and Operational Research, 20(4):394-422, 1982. URL:
  34. Ben Reichardt. Reflections for quantum query algorithms. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 560-569. SIAM, 2011. URL:
  35. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing exact minimum cuts without knowing the graph. In Proceedings of the 9th Innovations in Theoretical Computer Science Conference (ITCS), pages 39:1-39:16. LIPICS, 2018. URL:
  36. Robert E. Tarjan and Uzi Vishkin. Finding biconnected components and computing tree functions in logarithmic parallel time. In Proceedings of the 25th Annual Symposium on Foundations of Computer Science (FOCS), pages 12-20. IEEE, 1984. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail