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Quantum-Classical Equivalence for And-Functions

Authors Sreejata Kishor Bhattacharya , Farzan Byramji , Arkadev Chattopadhyay , Yogesh Dahiya , Shachar Lovett

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Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Quantum communication complexity
Keywords
  • Communication complexity
  • quantum communication complexity
  • De Morgan sparsity
  • approximate gamma two norm
  • And-functions

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Abstract

A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for And-functions of the form 

F(x,y) = f(x₁ ∧ y₁, …, x_n ∧ y_n), 

when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all And-functions has remained open and has been posed by several authors.
In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f∘And₂ are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized - up to polynomial loss - by the logarithm of the De Morgan sparsity of f.
Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett [Arkadev Chattopadhyay et al., 2026] on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general And-functions.

Cite As Get BibTex

Sreejata Kishor Bhattacharya, Farzan Byramji, Arkadev Chattopadhyay, Yogesh Dahiya, and Shachar Lovett. Quantum-Classical Equivalence for And-Functions. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.23

Author Details

Sreejata Kishor Bhattacharya
  • TIFR, Mumbai, India
Farzan Byramji
  • UC San Diego, La Jolla, CA, USA
Arkadev Chattopadhyay
  • TIFR, Mumbai, India
Yogesh Dahiya
  • UC San Diego, La Jolla, CA, USA
Shachar Lovett
  • UC San Diego, La Jolla, CA, USA

Funding

  • Bhattacharya, Sreejata Kishor: Supported by the Department of Atomic Energy, Govt. of India, under project #RTI4014 and by a Google PhD Fellowship.
  • Byramji, Farzan: Supported by Simons Investigator Award #929894, and NSF awards CCF-2425349 and AF: Medium 2212136.
  • Chattopadhyay, Arkadev: Supported by the Department of Atomic Energy, Govt. of India, under project #RTI4014 and by a Google India Faculty Award.
  • Dahiya, Yogesh: Supported by Simons Investigator Award #929894 and NSF award CCF-2425349.
  • Lovett, Shachar: Supported by Simons Investigator Award #929894 and NSF award CCF-2425349.

References

  1. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of huang’s sensitivity theorem. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1330-1342, 2021. URL: https://doi.org/10.1145/3406325.3451047.
  2. Andris Ambainis, Leonard J Schulman, Amnon Ta-Shma, Umesh Vazirani, and Avi Wigderson. The quantum communication complexity of sampling. SIAM Journal on Computing, 32(6):1570-1585, 2003. URL: https://doi.org/10.1137/S009753979935476.
  3. Nikhil Bansal and Makrand Sinha. K-forrelation optimally separates quantum and classical query complexity. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1303-1316, 2021. URL: https://doi.org/10.1145/3406325.3451040.
  4. Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Physical review letters, 87(16):167902, 2001. Google Scholar
  5. Harry Buhrman and Ronald de Wolf. Communication complexity lower bounds by polynomials. In Proceedings of the 16th Annual IEEE Conference on Computational Complexity, pages 120-130, 2001. URL: https://doi.org/10.1109/CCC.2001.933879.
  6. Harry Buhrman, Nikolai K. Vereshchagin, and Ronald de Wolf. On computation and communication with small bias. In 22nd Annual IEEE Conference on Computational Complexity (CCC 2007), 13-16 June 2007, San Diego, California, USA, pages 24-32. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/CCC.2007.18.
  7. Mark Bun and Justin Thaler. Approximate degree in classical and quantum computing. Found. Trends Theor. Comput. Sci., 15(3-4):229-423, 2022. URL: https://doi.org/10.1561/0400000107.
  8. Arkadev Chattopadhyay, Yogesh Dahiya, and Shachar Lovett. Restriction trees for sparsity and applications. In Proceedings of the 58th Annual ACM Symposium on Theory of Computing (STOC 2026), 2026. URL: https://eccc.weizmann.ac.il/report/2025/172/.
  9. Arkadev Chattopadhyay, Nikhil S. Mande, and Suhail Sherif. The log-approximate-rank conjecture is false. J.ACM, 67(4):1-28, 2020. URL: https://doi.org/10.1145/3396695.
  10. Richard Cleve and Harry Buhrman. Substituting quantum entanglement for communication. Physical Review A, 56(2):1201, 1997. Google Scholar
  11. Ronald de Wolf. Quantum communication and complexity. Theoretical computer science, 287(1):337-353, 2002. URL: https://doi.org/10.1016/S0304-3975(02)00377-8.
  12. Dmitry Gavinsky. Bare quantum simultaneity versus classical interactivity in communication complexity. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 401-411. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384243.
  13. Dmitry Gavinsky. Entangled simultaneity versus classical interactivity in communication complexity. IEEE Trans. Inf. Theory, 66(7):4641-4651, 2020. URL: https://doi.org/10.1109/TIT.2020.2976074.
  14. Dmitry Gavinsky, Julia Kempe, Oded Regev, and Ronald de Wolf. Bounded-error quantum state identification and exponential separations in communication complexity. In Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing, pages 594-603, 2006. URL: https://doi.org/10.1145/1132516.1132602.
  15. Uma Girish, Ran Raz, and Avishay Tal. Quantum versus randomized communication complexity, with efficient players. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, Virtual Conference, January 6-8, 2021, LIPIcs, pages 54:1-54:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.54.
  16. Mika Göös, Tom Gur, Siddhartha Jain, and Jiawei Li. Quantum communication advantage in TFNP. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, Prague, Czechia, June 23-27, 2025, pages 1465-1475. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718155.
  17. Bala Kalyanasundaram and Georg Schintger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. URL: https://doi.org/10.1137/0405044.
  18. Hartmut Klauck. Lower bounds for quantum communication complexity. SIAM J. Comput., 37(1):20-46, 2007. URL: https://doi.org/10.1137/S0097539702405620.
  19. Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan. Log-rank and lifting for and-functions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 197-208, 2021. URL: https://doi.org/10.1145/3406325.3450999.
  20. Matthias Krause and Pavel Pudlák. On computing boolean functions by sparse real polynomials. In 36th Annual Symposium on Foundations of Computer Science, FOCS 1995, Milwaukee, Wisconsin, USA, 23-25 October 1995, pages 682-691. IEEE Computer Society, 1995. URL: https://doi.org/10.1109/SFCS.1995.492670.
  21. Ilan Kremer. Quantum communication. Master’s Thesis, Hebrew University, Computer Science Department, 1995. Google Scholar
  22. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  23. Troy Lee and Adi Shraibman. An approximation algorithm for approximation rank. In 2009 24th Annual IEEE Conference on Computational Complexity, pages 351-357. IEEE, 2009. URL: https://doi.org/10.1109/CCC.2009.25.
  24. Troy Lee and Adi Shraibman. Lower bounds in communication complexity. Foundations and Trends® in Theoretical Computer Science, 3(4):263-399, 2009. URL: https://doi.org/10.1561/0400000040.
  25. Nati Linial and Adi Shraibman. Lower bounds in communication complexity based on factorization norms. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 699-708, 2007. URL: https://doi.org/10.1145/1250790.1250892.
  26. Ashley Montanaro and Tobias Osborne. On the communication complexity of XOR functions. arXiv preprint arXiv:0909.3392, 2009. URL: https://arxiv.org/abs/0909.3392.
  27. Noam Nisan. CREW prams and decision trees. SIAM J. Comput., 20(6):999-1007, 1991. URL: https://doi.org/10.1137/0220062.
  28. Anup Rao and Amir Yehudayoff. Communication complexity: and applications. Cambridge University Press, 2020. Google Scholar
  29. Ran Raz. Exponential separation of quantum and classical communication complexity. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 358-367, 1999. URL: https://doi.org/10.1145/301250.301343.
  30. Alexander A Razborov. On the distributional complexity of disjointness. In International Colloquium on Automata, Languages, and Programming, pages 249-253. Springer, 1990. URL: https://doi.org/10.1007/BFB0032036.
  31. Alexander A Razborov. Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics, 67(1):145, 2003. Google Scholar
  32. Oded Regev and Bo'az Klartag. Quantum one-way communication can be exponentially stronger than classical communication. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 31-40. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993642.
  33. Norbert Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, 1972. URL: https://doi.org/10.1016/0097-3165(72)90019-2.
  34. Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41(1):247-261, 1972. Google Scholar
  35. Alexander A Sherstov. The pattern matrix method for lower bounds on quantum communication. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 85-94, 2008. URL: https://doi.org/10.1145/1374376.1374392.
  36. Alexander A Sherstov. The pattern matrix method (journal version). arXiv preprint arXiv:0906.4291, 2009. URL: https://arxiv.org/abs/0906.4291.
  37. Alexander A. Sherstov. On quantum-classical equivalence for composed communication problems. Quantum Inf. Comput., 10(5&6):435-455, 2010. URL: https://doi.org/10.26421/QIC10.5-6-5.
  38. Alexander A. Sherstov. The multiparty communication complexity of set disjointness. SIAM J. Comput., 45(4):1450-1489, 2016. URL: https://doi.org/10.1137/120891587.
  39. Alexander A Sherstov, Andrey A Storozhenko, and Pei Wu. An optimal separation of randomized and quantum query complexity. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1289-1302, 2021. URL: https://doi.org/10.1145/3406325.3451019.
  40. Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. Quantum Inf. Comput., 9(5&6):444-460, 2009. URL: https://doi.org/10.26421/QIC9.5-6-7.
  41. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In Proceedings of the eleventh annual ACM symposium on Theory of computing, pages 209-213, 1979. URL: https://doi.org/10.1145/800135.804414.
  42. Andrew Chi-Chih Yao. Quantum circuit complexity. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 352-361. IEEE, 1993. URL: https://doi.org/10.1109/SFCS.1993.366852.
  43. Shengyu Zhang. On the tightness of the buhrman-cleve-wigderson simulation. In Yingfei Dong, Ding-Zhu Du, and Oscar Ibarra, editors, Algorithms and Computation, pages 434-440, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-10631-6_45.
  44. Zhiqiang Zhang and Yaoyun Shi. Communication complexities of symmetric XOR functions. Quantum information and computation, 9(3&4):255-263, 2009. URL: https://doi.org/10.26421/QIC9.3-4-5.
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