多项式的通信复杂度下界

H. Buhrman, R. D. Wolf
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引用次数: 134

摘要

通信复杂性的量子版本允许通信双方交换量子位和/或利用先验纠缠(共享epr对)。对于量子比特通信复杂性,已有一些下界技术,但除了内积函数外,对于具有无限先验纠缠的模型,没有已知的下界。我们证明了“对数秩”下界扩展到量子通信复杂性的最强变体(量子比特通信+无限先验纠缠)。通过将通信矩阵的秩与多项式的性质联系起来,我们能够推导出精确协议的一些强界。特别地,我们证明了各种函数的量子和经典通信复杂度的“对数秩猜想”和多项式等价性。我们还推导了有界误差量子协议的一些弱界。
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Communication complexity lower bounds by polynomials
The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.
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