具有共享输入的复合函数的量子算法和逼近多项式
Quantum algorithms and approximating polynomials for composed functions with shared inputs
摘要：
我们给出了新的量子算法，用于评估其输入可以在底层门之间共享的组合函数。令$f$是$m$位布尔函数，并考虑通过将$f$应用于$n$变量的可能重叠子集的合取而获得的$n$位函数$F$.如果$f$具有量子查询复杂度$Q(f)$，我们给出了一个利用$\tilde{O}(\sqrt{Q(f) \cdot n})$量子查询计算$F$的算法.这改进了$O(Q(f) \cdot \sqrt{n})$的边界，通过独立地处理每个连接，我们的边界对于$f$的最坏情况选择是紧的。使用完全不同的技术，我们证明了一个类似的紧复合定理的近似程度的$f$。通过递归地应用我们的复合定理，我们得到了一个近似最优的$\tilde{O}(n^{1-2^{-d}})$上界的量子查询复杂度和近似程度的线性尺寸深度- $d$ AC $^0$电路。因此，这样的电路可以在亚指数时间内学习PAC，即使在具有挑战性的不可知设置中。在我们的工作之前，一个次指数时间算法是不知道的，即使是线性尺寸的深度3 AC $^0$电路。作为一个额外的后果，我们表明，交流$^0 \circ \oplus$电路的深度$d+1$需要大小$\tildeΩ(n^{1/(1- 2^{-d})}) \geq ω(n^{1+ 2^{-d}} )$计算的内积函数，甚至平均。先前的最佳尺寸下限是$Ω(n^{1+4^{-(d+1)}})$并且仅在最差情况下保持（Cheraghchi等人，JCSS 2018）。
英文摘要：
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 $\tildeΩ(n^{1/(1- 2^{-d})}) \geq ω(n^{1+ 2^{-d}} )$ to compute the Inner Product function even on average. The previous best size lower bound was $Ω(n^{1+4^{-(d+1)}})$ and only held in the worst case (Cheraghchi et al., JCSS 2018).
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