Round Elimination in Exact Communication Complexity

Authors Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, Florian Speelman



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Jop Briët
Harry Buhrman
Debbie Leung
Teresa Piovesan
Florian Speelman

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Jop Briët, Harry Buhrman, Debbie Leung, Teresa Piovesan, and Florian Speelman. Round Elimination in Exact Communication Complexity. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 206-225, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.TQC.2015.206

Abstract

We study two basic graph parameters, the chromatic number and the orthogonal rank, in the context of classical and quantum exact communication complexity. In particular, we consider two types of communication problems that we call promise equality and list problems. For both of these, it was already known that the one-round classical and one-round quantum complexities are characterized by the chromatic number and orthogonal rank of a certain graph, respectively. In a promise equality problem, Alice and Bob must decide if their inputs are equal or not. We prove that classical protocols for such problems can always be reduced to one-round protocols with no extra communication. In contrast, we give an explicit instance of a promise problem that exhibits an exponential gap between the one- and two-round exact quantum communication complexities. Whereas the chromatic number thus captures the complete complexity of promise equality problems, the hierarchy of "quantum chromatic numbers" (starting with the orthogonal rank) giving the quantum communication complexity for every fixed number of communication rounds thus turns out to enjoy a much richer structure. In a list problem, Bob gets a subset of some finite universe, Alice gets an element from Bob's subset, and their goal is for Bob to learn which element Alice was given. The best general lower bound (due to Orlitsky) and upper bound (due to Naor, Orlitsky, and Shor) on the classical communication complexity of such problems differ only by a constant factor. We exhibit an example showing that, somewhat surprisingly, the four-round protocol used in the bound of Naor et al. can in fact be optimal. Finally, we pose a conjecture on the orthogonality rank of a certain graph whose truth would imply an intriguing impossibility of round elimination in quantum protocols for list problems, something that works trivially in the classical case.
Keywords
  • communication complexity
  • round elimination
  • quantum communication
  • protocols
  • chromatic numbers

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References

  1. Andris Ambainis. Quantum search algorithms. SIGACT News, 35(2):22-35, June 2004. Google Scholar
  2. M. Boyer, G. Brassard, P. Høyer, and A. Tapp. Tight bounds on quantum searching. Fortschritte der Physik, 46(4-5):493-505, 1998. Google Scholar
  3. G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. In Quantum computation and information (Washington, DC, 2000), volume 305 of Contemp. Math., pages 53-74. Amer. Math. Soc., Providence, RI, 2002. Google Scholar
  4. G. Brassard, P. Høyer, and A. Tapp. Quantum counting. In Proceedings of the 25th International Colloquium on Automata, Languages and Programming, ICALP'98, pages 820-831, London, UK, 1998. Springer-Verlag. Google Scholar
  5. J. Briët, H. Buhrman, and D. Gijswijt. Violating the Shannon capacity of metric graphs with entanglement. Proceedings of the National Academy of Sciences, 2012. Google Scholar
  6. J. Briët, H. Buhrman, M. Laurent, T. Piovesan, and G. Scarpa. Entanglement-assisted zero-error source-channel coding. Information Theory, IEEE Transactions on, 61(2):1124-1138, 2015. A previous version appeared in EuroComb 2014. Google Scholar
  7. H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC 1998), pages 63-68, 1998. Google Scholar
  8. P. J. Cameron, A. Montanaro, M. W. Newman, S. Severini, and A. Winter. On the quantum chromatic number of a graph. The electronic journal of combinatorics, 14(R81):1, 2007. Google Scholar
  9. T. S. Cubitt, D. Leung, W. Matthews, and A. Winter. Improving zero-error classical communication with entanglement. Physical Review Letters, 104(23):230503, 2010. Google Scholar
  10. T. S. Cubitt, L. Manc̆inska, D. E. Roberson, S. Severini, D. Stahlke, and A. Winter. Bounds on entanglement assisted source-channel coding via the lovasz theta number and its variants. IEEE Transactions of Information Theory, 60(11):7330-7344, 2014. Google Scholar
  11. R. de Wolf. Quantum Computing and Communication Complexity. PhD thesis, Universiteit van Amsterdam, 2001. Google Scholar
  12. P. Delsarte and V. I. Levenshtein. Association schemes and coding theory. IEEE Transactions on Information Theory, 44(6):2477-2504, 1998. Google Scholar
  13. A. G. Dýachkov and V. V. Rykov. Bounds on the length of disjunctive codes. Problemy Peredachi Informatsii, 18(3):7-13, 1982. In Russian. Google Scholar
  14. P. Frankl and V. Rödl. Forbidden intersections. Transactions of the American Mathematical Society, 300(1):259-286, 1987. Google Scholar
  15. Z. Füredi. On r-cover-free families. Journal of Combinatorial Theory, Series A, 73(1):172-173, 1996. Google Scholar
  16. C. D. Godsil and M. W. Newman. Coloring an orthogonality graph. SIAM Journal on Discrete Mathematics, 22(2):683-692, March 2008. Google Scholar
  17. L. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of STOC'96, pages 212-219, 1996. Google Scholar
  18. J. Gruska, D. Qiu, and S. Zheng. Generalizations of the distributed deutsch-jozsa promise problem. Preprint available at arXiv:1402.7254, 2014. Google Scholar
  19. A. S. Holevo. Bounds for the quantity of information transmitted by a quantum communication channel. Problems of Information Transmission, 9:177-183, 1973. Google Scholar
  20. I. Krasikov and S. Litsyn. Survey of binary Krawtchouk polynomials., pages 199-211. Providence, RI: AMS, American Mathematical Society, 2001. Google Scholar
  21. I. Kremer. Quantum communication. Master’s thesis, Hebrew University, Computer Science Department, 1995. Google Scholar
  22. E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  23. M. Laurent. Private communications, 2014. Google Scholar
  24. D. Leung, L. Manc̆inska, W. Matthews, M. Ozols, and A. Roy. Entanglement can increase asymptotic rates of zero-error classical communication over classical channels. Communications in Mathematical Physics, 311:97-111, 2012. Google Scholar
  25. V. I. Levenshtein. Krawtchouk polynomials and universal bounds for codes and designs in hamming spaces. IEEE Trans. Inf. Theor., 41(5):1303-1321, 1995. Google Scholar
  26. L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, 25(1):1-7, 1979. Google Scholar
  27. M. Naor, A. Orlitsky, and P. Shor. Three results on interactive communication. IEEE Transactions on Information Theory, 39:1608-1615, 1993. Google Scholar
  28. A. Orlitsky. Worst-case interactive communication i: Two messages are almost optimal. IEEE Transactions on Information Theory, 36:1111-1126, 1990. Google Scholar
  29. M. J. P. Peeters. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica, 16(3):417-431, 1996. Google Scholar
  30. M. Ruszinkó. On the upper bound of the size of the r-cover-free families. Journal of Combinatorial Theory, Series A, 66(2):302-310, 1994. Google Scholar
  31. G. Scarpa and S. Severini. Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number. IEEE Transactions on Information Theory, 58(4):2524-2529, 2012. Google Scholar
  32. Joel Spencer. Asymptopia, volume 71 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2014. With Laura Florescu. Google Scholar
  33. H. S. Witsenhausen. The zero-error side information problem and chromatic numbers. IEEE Transactions on Information Theory, 22(5):592-593, 1976. Google Scholar
  34. A. C.-C. Yao. Some complexity questions related to distributive computing (preliminary report). In Proceedings of the 11th annual ACM symposium on Theory of computing (STOC 1979), pages 209-213, 1979. Google Scholar
  35. A. C.-C. Yao. Quantum circuit complexity. In Proceedings of the 34th Annual Symposium on Foundations of Computer Science (FOCS 1993), pages 352-361, 1993. Google Scholar
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