Document Open Access Logo

Quantum Property Testing in Sparse Directed Graphs

Authors Simon Apers , Frédéric Magniez , Sayantan Sen , Dániel Szabó

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.32.pdf
  • Filesize: 0.83 MB
  • 24 pages
HTML5 Logo HTML (experimental)
LIPIcs.APPROX-RANDOM.2025.32.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Quantum query complexity
Keywords
  • property testing
  • quantum computing
  • bounded-degree directed graphs
  • dual polynomial method
  • collision finding

Metrics

Abstract

We initiate the study of quantum property testing in sparse directed graphs, and more particularly in the unidirectional model, where the algorithm is allowed to query only the outgoing edges of a vertex. In the classical unidirectional model, the problem of testing k-star-freeness, and more generally k-source-subgraph-freeness, is almost maximally hard for large k. We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we show that this advantage is nearly tight, by showing a quantum lower bound using the method of dual polynomials on an intermediate problem for a new, property testing version of the k-collision problem that was not studied before.
To illustrate that not all problems in graph property testing admit such a quantum speedup, we consider the problem of 3-colorability in the related undirected bounded-degree model, when graphs are now undirected. This problem is maximally hard to test classically, and we show that also quantumly it requires a linear number of queries.

Cite As Get BibTex

Simon Apers, Frédéric Magniez, Sayantan Sen, and Dániel Szabó. Quantum Property Testing in Sparse Directed Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 32:1-32:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.32

Author Details

Simon Apers
  • Université Paris Cité, CNRS, IRIF, Paris, France
Frédéric Magniez
  • Université Paris Cité, CNRS, IRIF, Paris, France
Sayantan Sen
  • Centre for Quantum Technologies, National University of Singapore, Singapore
Dániel Szabó
  • Université Paris Cité, CNRS, IRIF, Paris, France

Funding

Research supported in part by the ERC Advanced Grant PARQ (grant agreement No 885394), 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), the French ANR project QUOPS (ANR-22-CE47-0003-01), the National Research Foundation, Singapore through the National Quantum Office, hosted in A*STAR, under its Centre for Quantum Technologies Funding Initiative (S24Q2d0009), and the CQT Young Researcher Career Development Grant.

Acknowledgements

The authors would like to thank the anonymous reviewers for their comments which improved the presentation of the paper.

References

  1. Scott Aaronson. Quantum lower bound for the collision problem. In Proceedings of 34th Annual ACM Symposium on Theory of Computing (STOC), pages 635-642. Association for Computing Machinery, 2002. URL: https://doi.org/10.1145/509907.509999.
  2. Scott Aaronson, Andris Ambainis, Janis Iraids, Martins Kokainis, and Juris Smotrovs. Polynomials, quantum query complexity, and Grothendieck’s inequality. In Proceedings of the 31st Conference on Computational Complexity (CCC), volume 50, pages 25:1-25:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.25.
  3. Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. Journal of the ACM, 51(4):595-605, 2004. URL: https://doi.org/10.1145/1008731.1008735.
  4. A. Ambainis. Quantum search algorithms. SIGACT News, 35(2):22-35, 2004. URL: https://doi.org/10.1145/992287.992296.
  5. Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing, 1(3):37-46, 2005. URL: https://doi.org/10.4086/toc.2005.v001a003.
  6. Andris Ambainis, Aleksandrs Belovs, Oded Regev, and Ronald de Wolf. Efficient quantum algorithms for (gapped) group testing and junta testing. In Proceedings of the 27th annual Symposium on Discrete Algorithms (SODA), pages 903-922, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch65.
  7. Andris Ambainis, Andrew M Childs, and Yi-Kai Liu. Quantum property testing for bounded-degree graphs. In Proceedings of the 14th International Workshop and 15th International Conference on Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX-RANDOM), pages 365-376. Springer Berlin Heidelberg, 2011. URL: https://doi.org/10.5555/2033252.2033285.
  8. Simon Apers and Ronald De Wolf. Quantum speedup for graph sparsification, cut approximation, and Laplacian solving. SIAM Journal on Computing, 51(6):1703-1742, 2022. URL: https://doi.org/10.1137/21M1391018.
  9. Simon Apers, Frédéric Magniez, Sayantan Sen, and Dániel Szabó. Quantum property testing in sparse directed graphs. arXiv:2410.05001, 2024. https://doi.org/10.48550/arXiv.2410.05001.
  10. Simon Apers and Alain Sarlette. Quantum fast-forwarding: Markov chains and graph property testing. Quantum Information & Computation, 19(3–4):181-213, 2019. URL: https://doi.org/10.5555/3370245.3370246.
  11. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  12. Shalev Ben-David, Andrew M Childs, András Gilyén, William Kretschmer, Supartha Podder, and Daochen Wang. Symmetries, graph properties, and quantum speedups. In 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 649-660. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00066.
  13. Michael A Bender and Dana Ron. Testing properties of directed graphs: acyclicity and connectivity. Random Structures & Algorithms, 20(2):184-205, 2002. URL: https://doi.org/10.1002/rsa.10023.
  14. Arnab Bhattacharyya and Yuichi Yoshida. Property Testing - Problems and Techniques. Springer Singapore, 1 edition, 2022. URL: https://doi.org/10.1007/978-981-16-8622-1.
  15. Andrej Bogdanov, Kenji Obata, and Luca Trevisan. A lower bound for testing 3-colorability in bounded-degree graphs. In 43rd Annual Symposium on Foundations of Computer Science (FOCS), pages 93-102. IEEE Computer Society, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181886.
  16. Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp. Tight Bounds on Quantum Searching, chapter 10, pages 187-199. John Wiley & Sons, Ltd, 1999. URL: https://doi.org/10.1002/3527603093.ch10.
  17. Gilles Brassard, Peter Høyer, and Alain Tapp. Quantum cryptanalysis of hash and claw-free functions. In Proceedings of the 3rd Latin American Symposium on Theoretical Informatics (LATIN), pages 163-169. Springer Berlin Heidelberg, 1998. URL: https://doi.org/10.1007/BFb0054319.
  18. Harry Buhrman, Lance Fortnow, Ilan Newman, and Hein Röhrig. Quantum property testing. SIAM Journal on Computing, 37(5):1387-1400, 2008. URL: https://doi.org/10.1137/S0097539704442416.
  19. Mark Bun, Robin Kothari, and Justin Thaler. The polynomial method strikes back: Tight quantum query bounds via dual polynomials. Theory of Computing, 16(10):1-71, 2020. URL: https://doi.org/10.4086/toc.2020.v016a010.
  20. Mark Bun and Justin Thaler. A nearly optimal lower bound on the approximate degree of AC⁰. SIAM Journal on Computing, 49(4):FOCS17-59-FOCS17-96, 2020. URL: https://doi.org/10.1137/17M1161737.
  21. Hubie Chen and Yuichi Yoshida. Testability of homomorphism inadmissibility: Property testing meets database theory. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS), pages 365-382. Association for Computing Machinery, 2019. URL: https://doi.org/10.1145/3294052.3319679.
  22. Artur Czumaj, Pan Peng, and Christian Sohler. Relating two property testing models for bounded degree directed graphs. In Proceedings of the 48th annual Symposium on Theory of Computing (STOC), pages 1033-1045. Association for Computing Machinery, 2016. URL: https://doi.org/10.1145/2897518.2897575.
  23. Artur Czumaj and Christian Sohler. Sublinear-time algorithms. In Oded Goldreich, editor, Property Testing: Current Research and Surveys, pages 41-64. Springer Berlin Heidelberg, 2010. URL: https://doi.org/10.1007/978-3-642-16367-8_5.
  24. Eldar Fischer. The art of uninformed decisions. Bulletin of the EATCS, 75:97, 2001. Google Scholar
  25. Sebastian Forster, Danupon Nanongkai, Liu Yang, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2046-2065. Society for Industrial and Applied Mathematics, 2020. URL: https://doi.org/10.5555/3381089.3381215.
  26. Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781108135252.
  27. Oded Goldreich. On Testing Hamiltonicity in the Bounded Degree Graph Model, pages 282-292. Springer Nature Switzerland, 2025. URL: https://doi.org/10.1007/978-3-031-88946-2_15.
  28. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653-750, 1998. URL: https://doi.org/10.1145/285055.285060.
  29. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. URL: https://doi.org/10.1007/s00453-001-0078-7.
  30. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC), pages 212-219. Association for Computing Machinery, 1996. URL: https://doi.org/10.1145/237814.237866.
  31. Aram W Harrow, Cedric Yen-Yu Lin, and Ashley Montanaro. Sequential measurements, disturbance and property testing. In Proceedings of the 28th Annual Symposium on Discrete Algorithms (SODA), pages 1598-1611. Society for Industrial and Applied Mathematics, 2017. URL: https://doi.org/10.5555/3039686.3039791.
  32. Frank Hellweg and Christian Sohler. Property testing in sparse directed graphs: Strong connectivity and subgraph-freeness. In 20th Annual European Symposium on Algorithms (ESA), pages 599-610. Springer Berlin Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-33090-2_52.
  33. Troy Lee. A note on the sign degree of formulas. arXiv:0909.4607, 2009. URL: https://doi.org/10.48550/arXiv.0909.4607.
  34. Qipeng Liu and Mark Zhandry. On finding quantum multi-collisions. In Advances in Cryptology (EUROCRYPT), pages 189-218. Springer International Publishing, 2019. URL: https://doi.org/10.1007/978-3-030-17659-4_7.
  35. Nikhil S. Mande, Justin Thaler, and Shuchen Zhu. Improved approximate degree bounds for k-distinctness. In Proceedings of the 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC), volume 158, pages 2:1-2:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.TQC.2020.2.
  36. Ashley Montanaro and Ronald de Wolf. A Survey of Quantum Property Testing. Number 7 in Graduate Surveys. Theory of Computing Library, 2016. URL: https://doi.org/10.4086/toc.gs.2016.007.
  37. Yaron Orenstein and Dana Ron. Testing Eulerianity and connectivity in directed sparse graphs. Theoretical Computer Science, 412(45):6390-6408, 2011. URL: https://doi.org/10.1016/j.tcs.2011.06.038.
  38. Pan Peng and Yuyang Wang. An optimal separation between two property testing models for bounded degree directed graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP), volume 261, pages 96:1-96:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.96.
  39. Sofya Raskhodnikova, Dana Ron, Amir Shpilka, and Adam Smith. Strong lower bounds for approximating distribution support size and the distinct elements problem. SIAM Journal on Computing, 39(3):813-842, 2009. URL: https://doi.org/10.1137/070701649.
  40. Dana Ron. Algorithmic and analysis techniques in property testing. Foundations and Trends in Theoretical Computer Science, 5(2):73-205, 2010. URL: https://doi.org/10.1561/0400000029.
  41. Ronitt Rubinfeld and Asaf Shapira. Sublinear time algorithms. SIAM Journal on Discrete Mathematics, 25(4):1562-1588, 2011. URL: https://doi.org/10.1137/100791075.
  42. Alexander A. Sherstov. The pattern matrix method. SIAM Journal on Computing (SICOMP), 40(6):1969-2000, 2011. URL: https://doi.org/10.1137/080733644.
  43. Alexander A. Sherstov. The intersection of two halfspaces has high threshold degree. SIAM Journal on Computing, 42(6):2329-2374, 2013. URL: https://doi.org/10.1137/100785260.
  44. Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. Quantum Information & Computation, 9(5):444-460, 2009. URL: https://doi.org/10.5555/2011791.2011798.
  45. Yuichi Yoshida and Hiro Ito. Query-number preserving reductions and linear lower bounds for testing. IEICE transactions on Information and Systems, E93.D(2):233-240, 2010. URL: https://doi.org/10.1587/transinf.E93.D.233.
  46. Yuichi Yoshida and Hiro Ito. Testing k-edge-connectivity of digraphs. Journal of Systems Science and Complexity, 23(1):91-101, 2010. URL: https://doi.org/10.1007/s11424-010-9280-5.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact