Interaction in Quantum Communication and the Complexity of Set Disjointness

Conference Proceedings of the Annual ACM Symposium on Theory of Computing 10/2001; DOI: 10.1145/380752.380786
Source: DBLP


One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structure|they involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a simpler" quantum protocol|one that has similar eciency, but uses fewer message exchanges.

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Available from: Hartmut Klauck, Feb 28, 2013
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    • "Our starting point is a bounded-round lower bound for the traditional two-player communication problem TPJ k,t described above, where a " round " consists of one message from either Alice or Bob. This bound can be deduced from the work of Klauck et al. [27], who in fact studied the problem in the more general quantum communication setting. The underlying intuition is that of round elimination à la Miltersen et al. [30] and Sen [34]. "

    Preview · Article · Jan 2011
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    • "Using Theorem 1.1, we give a new and simple proof of Razborov's result. No alternate proof was available prior to this work, despite the fact that this problem has drawn the attention of various researchers [3] [12] [31] [29] [24] [42]. Moreover, the next-best lower bounds for general predicates were nowhere close to Theorem 1.3. "
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    ABSTRACT: We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f : {0,1}n ! {0,1} and let Af be the matrix whose columns are each an application of f to some subset of the variables x1,x2,...,x4n. We prove that Af has bounded-error communication complexity ( d), where d is the approximate degree of f. This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of Af in terms of well-studied analytic properties of f, broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has recently enabled important progress in multiparty communication complexity.
    Preview · Article · Jan 2011 · SIAM Journal on Computing
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    • "This is because many of these lower bound proofs use some notion of " self-reducibility, " arising from the original data structure problem, which fails to hold in the 'average case' but holds for the 'worst case.' The quantum round reduction arguments of Klauck et al. [22] are 'average case' arguments, and this is one of the reasons why they do not suffice to prove lower bounds for the rounds complexity of communication games arising from data structure problems. (2) For [t; c; l 1 , . . . "
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    Preview · Article · May 2008 · Journal of Computer and System Sciences
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