Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

Authors Olivier Lalonde, Nikhil S. Mande , Ronald de Wolf

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Olivier Lalonde
  • DIRO, Université de Montréal, Canada
Nikhil S. Mande
  • University of Liverpool, UK
Ronald de Wolf
  • QuSoft, CWI and University of Amsterdam, The Netherlands


We thank Troy Lee, Ignacio Villanueva, and Zhaohui Wei for early discussions related to the result of Section 4.3. We thank Swagato Sanyal for discussions at an early stage of this work, from which the question of pinning down the exact communication complexity of Equality for small error arose.

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Olivier Lalonde, Nikhil S. Mande, and Ronald de Wolf. Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability ε, getting the optimal constant factors in the leading terms in various different models. The following are our results in the randomized model: - We give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error at a small additive cost. This is an improvement over Newman’s theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. - As a consequence we obtain a (log(n/ε²) + 4)-cost private-coin communication protocol that computes the n-bit Equality function, to error ε. This improves upon the log(n/ε³) + O(1) upper bound implied by Newman’s theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive log log(1/ε) + O(1). The following are our results in various quantum models: - We exhibit a one-way protocol with log(n/ε) + 4 qubits of communication for the n-bit Equality function, to error ε, that uses only pure states. This bound was implicitly already shown by Nayak [PhD thesis'99]. - We give a near-matching lower bound: any ε-error one-way protocol for n-bit Equality that uses only pure states communicates at least log(n/ε) - log log(1/ε) - O(1) qubits. - We exhibit a one-way protocol with log(√n/ε) + 3 qubits of communication that uses mixed states. This is tight up to additive log log(1/ε) + O(1), which follows from Alon’s result. - We exhibit a one-way entanglement-assisted protocol achieving error probability ε with ⌈log(1/ε)⌉ + 1 classical bits of communication and ⌈log(√n/ε)⌉ + 4 shared EPR-pairs between Alice and Bob. This matches the communication cost of the classical public coin protocol achieving the same error probability while improving upon the amount of prior entanglement that is needed for this protocol, which is ⌈log(n/ε)⌉ + O(1) shared EPR-pairs. Our upper bounds also yield upper bounds on the approximate rank, approximate nonnegative-rank, and approximate psd-rank of the Identity matrix. As a consequence we also obtain improved upper bounds on these measures for a function that was recently used to refute the randomized and quantum versions of the log-rank conjecture (Chattopadhyay, Mande and Sherif [J. ACM'20], Sinha and de Wolf [FOCS'19], Anshu, Boddu and Touchette [FOCS'19]).

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ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Quantum complexity theory
  • Communication complexity
  • quantum communication complexity


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  1. Noga Alon. Perturbed identity matrices have high rank: Proof and applications. Combinatorics, Probability and Computing, 18(1-2):3-15, 2009. URL: https://doi.org/10.1017/S0963548307008917.
  2. Andris Ambainis. Communication complexity in a 3-computer model. Algorithmica, 16(3):298-301, 1996. Google Scholar
  3. Anurag Anshu, Naresh Goud Boddu, and Dave Touchette. Quantum log-approximate-rank conjecture is also false. In Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 982-994, 2019. URL: https://doi.org/10.1109/FOCS.2019.00063.
  4. Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Physical Review Letters, 87(16), September 26, 2001. URL: https://arxiv.org/abs/quant-ph/0102001.
  5. Harry Buhrman and Ronald de Wolf. Communication complexity lower bounds by polynomials. In Proceedings of the 16th Annual IEEE Conference on Computational Complexity (CCC), pages 120-130, 2001. URL: https://doi.org/10.1109/CCC.2001.933879.
  6. Arkadev Chattopadhyay, Nikhil S. Mande, and Suhail Sherif. The log-approximate-rank conjecture is false. Journal of the ACM, 67(4):23:1-23:28, 2020. Earlier version in STOC, 2019. URL: https://dl.acm.org/doi/10.1145/3396695.
  7. Hamza Fawzi, João Gouveia, Pablo A. Parrilo, Richard Z. Robinson, and Rekha R. Thomas. Positive semidefinite rank. Mathematical Programming, 153(1):133-177, 2015. URL: https://doi.org/10.1007/s10107-015-0922-1.
  8. Patrick Hayden, Debbie W. Leung, and Andreas Winter. Aspects of generic entanglement. Communications in Mathematical Physics, 265(1):95-117, 2006. Google Scholar
  9. Matthias Krause. Geometric arguments yield better bounds for threshold circuits and distributed computing. Theoretical Computer Science, 156(1&2):99-117, 1996. URL: https://doi.org/10.1016/0304-3975(95)00005-4.
  10. Ilan Kremer. Quantum communication. Master’s thesis, Hebrew University, Computer Science Department, 1995. Google Scholar
  11. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  12. Troy Lee, Zhaohui Wei, and Ronald de Wolf. Some upper and lower bounds on psd-rank. Mathematical Programming, Series A, 162(1-2):495-521, 2017. Google Scholar
  13. Elizabeth Meckes. Concentration of measure and the compact classical matrix groups, 2014. Lecture notes, available at https://case.edu/artsci/math/esmeckes/Haar_-notes.pdf. Google Scholar
  14. Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005. URL: https://doi.org/10.1017/CBO9780511813603.
  15. Ashwin Nayak. Lower bounds for Quantum Computation and Communication. PhD thesis, University of California, Berkeley, 1999. Google Scholar
  16. Ilan Newman. Private vs. common random bits in communication complexity. Information Processing Letters, 39(2):67-71, 1991. URL: https://doi.org/10.1016/0020-0190(91)90157-D.
  17. Ilan Newman and Mario Szegedy. Public vs. private coin flips in one round communication games. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC), pages 561-570, 1996. Google Scholar
  18. Michael A. Nielsen. Quantum Information Theory. PhD thesis, University of New Mexico, Albuquerque, 1998. Google Scholar
  19. Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. Google Scholar
  20. Anup Rao and Amir Yehudayoff. Communication Complexity and Applications. Cambridge University Press, 2020. Google Scholar
  21. Makrand Sinha and Ronald de Wolf. Exponential separation between quantum communication and logarithm of approximate rank. In Proceedings of the 60th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 966-981, 2019. URL: https://doi.org/10.1109/FOCS.2019.00062.
  22. Andreas Winter. Quantum and classical message identification via quantum channels. In O. Hirota, editor, Festschrift A.S. Holevo 60. Rinton, 2004. URL: https://arxiv.org/abs/quant-ph/0401060.
  23. 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.
  24. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In Proceedings of the 11h Annual ACM Symposium on Theory of Computing (STOC), pages 209-213, 1979. Google Scholar
  25. Andrew Chi-Chih Yao. Quantum circuit complexity. In Proceedings of the 34th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 352-361, 1993. Google Scholar
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