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

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Book cover Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6755))

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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.

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Authors and Affiliations

  1. The Chinese University of Hong Kong, Hong Kong

    Shengyu Zhang

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  1. Shengyu Zhang
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Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

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Zhang, S. (2011). On the Power of Lower Bound Methods for One-Way Quantum Communication Complexity. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_5

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  • DOI: https://doi.org/10.1007/978-3-642-22006-7_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22005-0

  • Online ISBN: 978-3-642-22006-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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