SIAM Journal on Computing, Ahead of Print.
The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \mathrm{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-\Omega(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $\delta > 0$ we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves upon the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi [J. ACM, 51 (2004), pp. 595--605] and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) disjunctive normal forms of polylogarithmic width. We describe several applications of these results and provide the following: (i) for any constant $\delta > 0$, an $\Omega(n^{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$^0$; (ii) a Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$; this separation is optimal up to the $o(1)$ term in the exponent; (iii) improved secret sharing schemes with reconstruction procedures in AC$^0$.

中文翻译：

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AC $ ^ 0 $的近似度的近似最佳下界

《 SIAM计算杂志》，预印本。
布尔函数$ f \ colon \ {-1，1 \} ^ n \ rightarrow \ {-1，1 \} $的近似度是实多项式的最小度，它最多将$ f $逐点近似到误差1/3。我们介绍了一种通用方法，用于增加给定函数的近似度，同时通过恒定深度电路保持其可计算性。具体来说，我们展示如何将近似度为$ d $的布尔函数$ f $转换为近似度至少为$ D的$ O（n \ cdot \ mathrm {polylog}（n））$变量上的函数$ F $ = \ Omega（n ^ {1/3} \ cdot d ^ {2/3}）$。特别是，如果$ d = n ^ {1- \ Omega（1）} $，则$ D $比$ d $多项式大。此外，如果$ f $是由恒定深度的多项式大小的布尔电路计算的，则$ F $也是如此。通过递归地应用我们的变换，对于任何常量$ \ delta> 0 $表现出AC $ ^ 0 $函数的近似度$ \ Omega（n ^ {1- \ delta}）$。这是由于Aaronson和Shi所致的$ \ Omega（n ^ {2/3}）$的最佳先前下界。ACM，51（2004），第595--605页]，几乎与任何函数都具有的$ n $的平凡上限相匹配。我们的下限也适用于（对数多项式大小）多对数宽度的析取正态形式。我们描述了这些结果的几种应用，并提供了以下内容：（i）对于任何常数$ \ delta> 0 $，函数量子通信复杂度的$ \ Omega（n ^ {1- \ delta}）$下界以AC $ ^ 0 $; （ii）布尔函数$ f $，其近似度至少为$ C（f）^ {2-o（1）} $，其中$ C（f）$是证书复杂度$ f $；在指数的$ o（1）$项之前，这种分离是最佳的；