SIAM Journal on Computing, Volume 49, Issue 6, Page 1173-1231, January 2020.
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a novel, first-principles approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: $O(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}})$ for the $k$-element distinctness problem; $O(n^{1-\frac{1}{k+1}})$ for the $k$-subset sum problem; $O(n^{1-\frac{1}{k+1}})$ for any $k$-DNF or $k$-CNF formula; and $O(n^{3/4})$ for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the $\Theta(n)$ quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree $\Omega(n)$. In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.

中文翻译：

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算法多项式

SIAM计算学报，第49卷，第6期，第1173-1231页，2020年1月。
布尔函数$ f（x_ {1}，x_ {2}，\ ldots，x_ {n}）$的近似度是实多项式的最小度，在$ 1/3 $内逐点近似$ f $。一般而言，近似度的上限在学习理论，差分隐私和算法设计中有多种应用。近似度的几乎所有已知上限都以存在方式从量子查询复杂度的界线中产生。我们为布尔函数的多项式逼近开发了一种新颖的第一原理方法。我们使用它为几个基本问​​题的近似度给出第一个建设性的上限：$ O（n ^ {\ frac {3} {4}-\ frac {1} {4（2 ^ {k} -1） }}）$代表$ k $元素的唯一性问题；$ O（n ^ {1- \ frac {1} {k + 1}}）$用于$ k $子集和问题；对于任何$ k $ -DNF或$ k $ -CNF公式，为$ O（n ^ {1- \ frac {1} {k + 1}}）$；和$ O（n ^ {3/4}）$来解决外射性问题。在所有情况下，我们都获得了与先前工作的量子论点无关的显式，闭式近似多项式。我们的前三个结果与量子查询复杂性的界限相匹配。我们的第四个结果在问题的$ \ Theta（n）$量子查询复杂度上进行了多项式改进，并驳斥了几位专家的推测，即射度具有近似度\\ Omega（n）$。特别是，我们展示了第一个自然问题，其近似程度与量子查询复杂度之间存在多项式差距。我们的第四个结果在问题的$ \ Theta（n）$量子查询复杂度上进行了多项式改进，并驳斥了几位专家的推测，即射度具有近似度\\ Omega（n）$。特别是，我们展示了第一个自然问题，其近似程度与量子查询复杂度之间存在多项式差距。我们的第四个结果在问题的$ \ Theta（n）$量子查询复杂度上进行了多项式改进，并驳斥了几位专家的推测，即射度具有近似度\\ Omega（n）$。特别是，我们展示了第一个自然问题，其近似程度与量子查询复杂度之间存在多项式差距。