Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) is at most 2R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively. We show that the O(log n) overhead is required for some functions in the quantum setting, and thus the BCW simulation is tight. We note here that prior to our work, the possibility of Q^{cc}(f o G) = O(Q(f)), for all f and all G in {AND_2, XOR_2}, had not been ruled out. More specifically, we show the following. - We show that the log n overhead is *not* required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2. - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model.

翻译：布尔曼、克莱夫和维格德森（STOC'98）证明：对于任意布尔函数 f : {-1,1}^n → {-1,1} 及 G ∈ {AND_2, XOR_2}，复合函数 f o G 的有界误差量子通信复杂度等于 O(Q(f) log n)，其中 Q(f) 表示 f 的有界误差量子查询复杂度。该结论通过Alice运行f的最优量子查询算法实现，每轮查询需通过O(log n)量子比特通信完成。这与经典情形形成对比——经典情形下易证R^{cc}(f o G) ≤ 2R(f)（其中R^{cc}和R分别表示有界误差通信与查询复杂度）。我们证明对于某些函数，量子情形下O(log n)的代价是必要的，因此BCW模拟是紧的。需指出，在本文工作之前，对于所有f及所有G ∈ {AND_2, XOR_2}，可能性Q^{cc}(f o G) = O(Q(f))尚未被排除。具体而言，我们证明以下结果： - 当f为对称函数时，log n代价并非必需，该结果推广了Aaronson和Ambainis关于集合不相交函数（Theory of Computing'05）的结论。 - 为证明上述结论，我们设计了带噪幅度放大的高效分布式版本，从而在f为OR函数时证明该结果。 - 基于第一个结论，自然产生疑问：即使对于对称性更弱的传递函数，BCW模拟中的log n代价是否可避免？我们给出强烈否定答案：即使允许量子通信协议的误差概率任意接近1/2，某些传递函数仍需要log n代价。 - 此外，我们还给出了一般性方法，用于构造有界误差通信模型下BCW模拟中必需log n代价的函数。