We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of possibly overlapping subsets of $n$ variables. If $f$ has quantum query complexity $Q(f)$, we give an algorithm for evaluating $F$ using $\tilde{O}(\sqrt{Q(f) \cdot n})$ quantum queries. This improves on the bound of $O(Q(f) \cdot \sqrt{n})$ that follows by treating each conjunction independently, and our bound is tight for worst-case choices of $f$. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of $f$.
By recursively applying our composition theorems, we obtain a nearly optimal $\tilde{O}(n^{1-2^{-d}})$ upper bound on the quantum query complexity and approximate degree of linear-size depth-$d$ AC$^0$ circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC$^0$ circuits.
As an additional consequence, we show that AC$^0 \circ \oplus$ circuits of depth $d+1$ require size $\tilde{\Omega}(n^{1/(1- 2^{-d})}) \geq \omega(n^{1+ 2^{-d}} )$ to compute the Inner Product function even on average. The previous best size lower bound was $\Omega(n^{1+4^{-(d+1)}})$ and only held in the worst case (Cheraghchi et al., JCSS 2018).

中文翻译：

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具有共享输入的复合函数的量子算法和近似多项式

我们提供了新的量子算法来评估组合函数，其输入可以在底层门之间共享。让 $f$ 是一个 $m$-bit 布尔函数，并考虑通过将 $f$ 应用于 $n$ 变量的可能重叠子集的连接而获得的 $n$-bit 函数 $F$。如果 $f$ 具有量子查询复杂度 $Q(f)$，我们给出了使用 $\tilde{O}(\sqrt{Q(f) \cdot n})$ 量子查询来评估 $F$ 的算法。这改进了 $O(Q(f) \cdot \sqrt{n})$ 的边界，通过独立处理每个连词，我们的边界对于 $f$ 的最坏情况选择是严格的。使用完全不同的技术，我们证明了 $f$ 的近似度的类似紧合成定理。
通过递归地应用我们的组合定理，我们获得了量子查询复杂度和线性大小深度的近似程度的近似最优的 $\tilde{O}(n^{1-2^{-d}})$ 上界-$ d$ AC$^0$ 电路。因此，即使在具有挑战性的不可知环境中，也可以在次指数时间内学习此类电路。在我们的工作之前，即使对于线性大小的深度 3 AC$^0$ 电路，亚指数时间算法也是未知的。
作为额外的结果，我们表明深度为 $d+1$ 的 AC$^0 \circ \oplus$ 电路需要大小 $\tilde{\Omega}(n^{1/(1- 2^{-d}) }) \geq \omega(n^{1+ 2^{-d}} )$ 甚至平均计算内积函数。之前的最佳尺寸下限是 $\Omega(n^{1+4^{-(d+1)}})$ 并且只在最坏的情况下成立（Cheraghchi 等人，JCSS 2018）。