Advertisement

On the Power of Lower Bound Methods for One-Way Quantum Communication Complexity

  • Shengyu Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

One of the most fundamental questions in communication complexity is the largest gap between classical and quantum one-way communication complexities, and it is conjectured that they are polynomially related for all total Boolean functions f. One approach to proving the conjecture is to first show a quantum lower bound L(f), and then a classical upper bound U(f) = poly(L(f)). Note that for this approach to be possibly successful, the quantum lower bound L(f) has to be polynomially tight for all total Boolean functions f.

This paper studies all the three known lower bound methods for one-way quantum communication complexity, namely the Partition Tree method by Nayak, the Trace Distance method by Aaronson, and the two-way quantum communication complexity. We deny the possibility of using the aforementioned approach by any of these known quantum lower bounds, by showing that each of them can be at least exponentially weak for some total Boolean functions. In particular, for a large class of functions generated from Erdös-Rényi random graphs G(N,p), with p in some range of 1/poly(N), though the two-way quantum communication complexity is linear in the size of input, the other two methods (particularly for the one-way model) give only constant lower bounds. En route of the exploration, we also discovered that though Nayak’s original argument gives a lower bound by the VC-dimension, the power of its natural extension, the Partition Tree method, turns out to be exactly equal to another measure in learning theory called the extended equivalence query complexity.

Keywords

Random Graph Communication Complexity Common Neighbor Full Version Trace Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aaronson, S.: Limitations of quantum advice and one-way communication. Theory of Computing 1, 1–28 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aaronson, S.: The learnability of quantum states. Proceedings of the Royal Society A 463, 2088 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing 1, 47–79 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambainis, A., Nayak, A., Ta-Shma, A., Vazirani, U.: Dense quantum coding and quantum finite automata. Journal of the ACM 49(4), 1–16 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Angluin, D.: Queries revisited. Theoretical Computer Science 313(2), 175–194 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bar-Yossef, Z., Jayram, T.S., Kerenidis, I.: Exponential separation of quantum and classical one-way communication complexity. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 128–137 (2004)Google Scholar
  7. 7.
    Buhrman, H., Cleve, R., Watrous, J., de Wolf, R.: Quantum fingerprinting. Physical Review Letters 87(16) (2001)Google Scholar
  8. 8.
    Buhrman, H., Cleve, R., Wigderson, A.: Quantum vs. classical communication and computation. In: Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing (STOC), pp. 63–68 (1998)Google Scholar
  9. 9.
    Gavinsky, D.: Classical interaction cannot replace a quantum message. In: Proceedings of the Fortieth Annual ACM Symposium on the Theory of Computing (STOC), pp. 95–102 (2008)Google Scholar
  10. 10.
    Gavinsky, D., Kempe, J., Kerenidis, I., Raz, R., de Wolf, R.: Exponential separation of quantum and classical one-way communication complexity. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), pp. 516–525 (2007)Google Scholar
  11. 11.
    Gavinsky, D., Pudlák, P.: Exponential separation of quantum and classical non-interactive multi-party communication complexity. In: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, pp. 332–339 (2008)Google Scholar
  12. 12.
    Høyer, P., Mosca, M., de Wolf, R.: Quantum search on bounded-error inputs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 291–299. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Jain, R., Klauck, H., Nayak, A.: Direct product theorems for classical communication complexity via subdistribution bounds. In: Proceedings of the Fortieth Annual ACM Symposium on the Theory of Computing (STOC), pp. 599–608 (2008)Google Scholar
  14. 14.
    Jain, R., Zhang, S.: New bounds on classical and quantum one-way communication complexity. Theoretical Computer Science 410(26), 2463–2477 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kalyanasundaram, B., Schintger, G.: The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics 5(4), 545–557 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klartag, B., Regev, O.: Quantum one-way communication can be exponentially stronger than classical communication. In: Proceedings of the 44th Annual ACM Symposium on the Theory of Computing (STOC) (to appear, 2011)Google Scholar
  17. 17.
    Klauck, H.: Quantum communication complexity. In: ICALP Satellite Workshops, pp. 241–252 (2000)Google Scholar
  18. 18.
    Klauck, H.: Lower bounds for quantum communication complexity. SIAM Journal on Computing 37(1), 20–46 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  20. 20.
    Linial, N., Shraibman, A.: Lower bounds in communication complexity based on factorization norms. In: Proceedings of the Thirty-Ninth Annual ACM symposium on Theory of Computing (STOC), pp. 699–708 (2007)Google Scholar
  21. 21.
    Littlestone, N.: Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning 2(4), 285–318 (1988)Google Scholar
  22. 22.
    Muthukrishnan, S.M.: Data streams: Algorithms and applications. Foundations and Trends in Theoretical Computer Science 1(2) (2005)Google Scholar
  23. 23.
    Nayak, A.: Optimal lower bounds for quantum automata and random access codes. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 124–133 (1999)Google Scholar
  24. 24.
    Raz, R.: Exponential separation of quantum and classical communication complexity. In: Proceedings of the 31st Annual ACM Symposium on the Theory of Computing (STOC), pp. 358–367 (1999)Google Scholar
  25. 25.
    Razborov, A.: On the distributional complexity of disjointness. Theoretical Computer Science 106, 385–390 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Razborov, A.: Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics 67(1), 145–159 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sherstov, A.: The pattern matrix method for lower bounds on quantum communication. In: Proceedings of the 40th Annual ACM Symposium on the Theory of Computing, pp. 85–94 (2008)Google Scholar
  28. 28.
    Wigderson, A.: Depth through breadth, or why should we attend talks in other areas? In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), p. 579 (2004), http://www.math.ias.edu/~avi/TALKS/STOC04.ppt
  29. 29.
    Winter, A.: Quantum and classical message identification via quantum channels. Quantum Information and Computation 4(6&7), 563–578 (2004)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Yao, A.C.-C.: Some complexity questions related to distributive computing. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing (STOC), pp. 209–213 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Shengyu Zhang
    • 1
  1. 1.The Chinese University of Hong KongHong Kong

Personalised recommendations