skip to main content
10.1145/1374376.1374392acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
research-article

The pattern matrix method for lower bounds on quantum communication

Published: 17 May 2008 Publication History

Abstract

In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.

References

[1]
S. Aaronson and Y. Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595--605, 2004.
[2]
A. Ambainis, L. J. Schulman, A. Ta--Shma, U. V. Vazirani, and A. Wigderson. The quantum communication complexity of sampling. SIAM J. Comput., 32(6):1570--1585, 2003.
[3]
R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778--797, 2001.
[4]
H. Buhrman and R. de Wolf. Communication complexity lower bounds by polynomials. In CCC, pages 120--130, 2001.
[5]
H. Buhrman, N. K. Vereshchagin, and R. de Wolf. On computation and communication with small bias. In CCC, pages 24--32, 2007.
[6]
A. Chattopadhyay. Discrepancy and the power of bottom fan-in in depth-three circuits. In FOCS, 2007.
[7]
A. Chattopadhyay and A. Ada. Multiparty communication complexity of disjointness. In Electronic Colloquium on Computational Complexity (ECCC), January 2008. Report TR08-002.
[8]
M. David and T. Pitassi. Separating NOF communication complexity classes RP and NP. In Electronic Colloquium on Computational Complexity (ECCC), February 2008. Report TR08-014.
[9]
R. de Wolf. Personal communication, October 2007.
[10]
R. A. DeVore and G. G. Lorentz. Constructive Approximation, volume 303. Springer-Verlag, Berlin, 1993.
[11]
D. Gavinsky, J. Kempe, and R. de Wolf. Strengths and weaknesses of quantum fingerprinting. In CCC, pages 288--298, 2006.
[12]
R. A. Horn and C. R. Johnson. Matrix analysis. New York, 1986.
[13]
P. Hoyer and R. de Wolf. Improved quantum communication complexity bounds for disjointness and equality. In STACS, pages 299--310, 2002.
[14]
A. D. Ioffe and V. M. Tikhomirov. Duality of convex functions and extremum problems. Russ. Math. Surv., 23(6):53--124, 1968.
[15]
H. Klauck. Lower bounds for quantum communication complexity. In FOCS, pages 288--297, 2001.
[16]
H. Klauck, A. Nayak, A. Ta-Shma, and D. Zuckerman. Interaction in quantum communication and the complexity of set disjointness. In STOC, pages 124--133, 2001.
[17]
I. Kremer. Quantum communication. Master's thesis, Hebrew University, Computer Science Department, 1995.
[18]
E. Kushilevitz and N. Nisan. Communication complexity. Cambridge University Press, New York, 1997.
[19]
T. Lee and A. Shraibman. Disjointness is hard in the multi-party number-on-the-forehead model. In CCC, 2008. To appear.
[20]
T. Lee, A. Shraibman, and R. Spalek. A direct product theorem for discrepancy. In CCC, 2008. To appear.
[21]
N. Linial and A. Shraibman. Lower bounds in communication complexity based on factorization norms. In STOC, pp. 699--708, 2007.
[22]
N. Linial and A. Shraibman. Learning complexity vs. communication complexity. In CCC, 2008. To appear.
[23]
M. L. Minsky and S. A. Papert. Perceptrons: expanded edition. MIT Press, Cambridge, Mass., 1988.
[24]
R. Paturi. On the degree of polynomials that approximate symmetric Boolean functions. In STOC, pages 468--474, 1992.
[25]
A. A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics, 67(1):145--159, 2003.
[26]
A. A. Razborov. Personal communication, June 2007.
[27]
T. J. Rivlin. An Introduction to the Approximation of Functions. Dover Publications, New York, 1981.
[28]
A. Schrijver. Theory of linear and integer programming. John Wiley & Sons, Inc., New York, 1998.
[29]
A. A. Sherstov. Separating AC0 from depth-2 majority circuits. In STOC, pages 294--301, 2007.
[30]
Y. Shi and Y. Zhu. Quantum communication complexity of block-composed functions. Available at arXiv:0710.0095v1, 29 September 2007.
[31]
A. C.-C. Yao. Quantum circuit complexity. In FOCS, pages 352--361, 1993.

Cited By

View all
  • (2019)Sensitivity, Affine Transforms and Quantum Communication ComplexityComputing and Combinatorics10.1007/978-3-030-26176-4_12(140-152)Online publication date: 21-Jul-2019
  • (2017)Exponential separation of quantum communication and classical informationProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055401(277-288)Online publication date: 19-Jun-2017
  • (2012)The multiparty communication complexity of set disjointnessProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214026(525-548)Online publication date: 19-May-2012
  • Show More Cited By

Index Terms

  1. The pattern matrix method for lower bounds on quantum communication

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Conferences
    STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
    May 2008
    712 pages
    ISBN:9781605580470
    DOI:10.1145/1374376
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

    Sponsors

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 17 May 2008

    Permissions

    Request permissions for this article.

    Check for updates

    Author Tags

    1. approximate degree of boolean functions
    2. bounded-error communication
    3. lower bounds
    4. quantum communication complexity

    Qualifiers

    • Research-article

    Conference

    STOC '08
    Sponsor:
    STOC '08: Symposium on Theory of Computing
    May 17 - 20, 2008
    British Columbia, Victoria, Canada

    Acceptance Rates

    STOC '08 Paper Acceptance Rate 80 of 325 submissions, 25%;
    Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

    Upcoming Conference

    STOC '25
    57th Annual ACM Symposium on Theory of Computing (STOC 2025)
    June 23 - 27, 2025
    Prague , Czech Republic

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)24
    • Downloads (Last 6 weeks)4
    Reflects downloads up to 20 Jan 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2019)Sensitivity, Affine Transforms and Quantum Communication ComplexityComputing and Combinatorics10.1007/978-3-030-26176-4_12(140-152)Online publication date: 21-Jul-2019
    • (2017)Exponential separation of quantum communication and classical informationProceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3055399.3055401(277-288)Online publication date: 19-Jun-2017
    • (2012)The multiparty communication complexity of set disjointnessProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214026(525-548)Online publication date: 19-May-2012
    • (2012)Information Complexity versus Corruption and Applications to Orthogonality and Gap-HammingApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques10.1007/978-3-642-32512-0_41(483-494)Online publication date: 2012
    • (2012)The NOF multiparty communication complexity of composed functionsProceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I10.1007/978-3-642-31594-7_2(13-24)Online publication date: 9-Jul-2012
    • (2012)The relationship between inner product and counting cyclesProceedings of the 10th Latin American international conference on Theoretical Informatics10.1007/978-3-642-29344-3_54(643-654)Online publication date: 16-Apr-2012
    • (2011)On the power of lower bound methods for one-way quantum communication complexityProceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I10.5555/2027127.2027134(49-60)Online publication date: 4-Jul-2011
    • (2011)An optimal lower bound on the communication complexity of gap-hamming-distanceProceedings of the forty-third annual ACM symposium on Theory of computing10.1145/1993636.1993644(51-60)Online publication date: 6-Jun-2011
    • (2011)On Arthur Merlin Games in Communication ComplexityProceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity10.1109/CCC.2011.33(189-199)Online publication date: 8-Jun-2011
    • (2011)On the Power of Lower Bound Methods for One-Way Quantum Communication ComplexityAutomata, Languages and Programming10.1007/978-3-642-22006-7_5(49-60)Online publication date: 2011
    • Show More Cited By

    View Options

    Login options

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Media

    Figures

    Other

    Tables

    Share

    Share

    Share this Publication link

    Share on social media