Consider the expected query complexity of computing the $k$-fold direct product $f^{\otimes k}$ of a function $f$ to error $\varepsilon$ with respect to a distribution $\mu^k$. One strategy is to sequentially compute each of the $k$ copies to error $\varepsilon/k$ with respect to $\mu$ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new "resilience lemma" that accompanies it, showing that the hardcore of $f^{\otimes k}$ is likely to remain dense under arbitrary partitions of the input space.

翻译：考虑计算函数$f$的$k$重直积$f^{\otimes k}$在分布$\mu^k$下达到误差$\varepsilon$的期望查询复杂度。一种策略是依次计算每个副本在分布$\mu$下达到误差$\varepsilon/k$，并应用联合界。我们证明了一个强直和定理，表明这种朴素策略本质上是最优的。具体而言，计算直积必然导致查询复杂度和误差的同时放大。强直和定理与那些仅显示查询复杂度或误差单方面放大的结果形成对比。在分布查询复杂度领域，此类结果可追溯至(Impagliazzo, Raz, Wigderson 1994)和(Nisan, Rudich, Saks 1994)的开创性工作，但强直和定理长期未能建立。我们研究中的关键思想是首次在查询复杂度背景下应用硬核定理(Impagliazzo 1995)。为此我们证明了一个新的"韧性引理"，表明$f^{\otimes k}$的硬核在输入空间的任意划分下很可能保持稠密性。