Strong direct product theorems for quantum communication and query complexity
Proceedings of the forty-third annual ACM symposium on Theory of computing
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The "direct product problem'' is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T$-query algorithmhas success probability at most 1 - \eps in computing the Boolean function f on input distribution mu, then for alpha \leq 1, every alpha \eps Tk-query algorithm has success probability at most (2^{\alpha \eps}(1-\eps))^k in computing the k-fold direct product f^{\otimes k} correctly on k independent inputs from \mu. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant $\alpha 0$, the worst-case success probability of any $\alpha R_2(f) k$-query randomized algorithm for f^{\otimes k} falls exponentially with k. The best previous statement of this type, due to Klauck, \v{S}palek, and de Wolf, required a query bound of O(bs(f) k). Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f^{\otimes k}. Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.