We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $\ell\geq1$ sum to at most $c^{\ell}\sqrt{\binom{d}{\ell}(1+\log n)^{\ell-1}},$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with $\ell,$ becoming trivial already at $\ell=\sqrt{d}.$ As an application, we obtain, for every integer $k\geq1,$ a partial Boolean function on $n$ bits that has bounded-error quantum query complexity at most $k$ and randomized query complexity $\tilde{\Omega}(n^{1-\frac{1}{2k}}).$ This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015) and Bravyi, Gosset, Grier, and Schaeffer (2021). Prior to our work, the best known separation was polynomially weaker: $O(1)$ versus $\Omega(n^{2/3-\epsilon})$ for any $\epsilon>0$ (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of $O(\log n)$ versus $\Omega(n^{1-\epsilon})$ for bounded-error quantum versus randomized communication complexity, for any $\epsilon>0.$ The best previous separation was polynomially weaker: $O(\log n)$ versus $\Omega(n^{2/3-\epsilon})$ (implicit in Tal, FOCS 2020).

翻译：我们证明，对于任意决策树，给定阶数 $\ell\geq1$ 的傅里叶系数绝对值之和至多为 $c^{\ell}\sqrt{\binom{d}{\ell}(1+\log n)^{\ell-1}}$，其中 $n$ 为变量数，$d$ 为树的深度，$c>0$ 为绝对常数。这一界本质上是紧的，并解决了 Tal（arXiv 2019；FOCS 2020）的一个猜想。我们之前的工作中的界随 $\ell$ 迅速退化，在 $\ell=\sqrt{d}$ 时已变得平凡。作为应用，我们得到，对于每个整数 $k\geq1$，存在一个定义在 $n$ 比特上的部分布尔函数，其有界错误量子查询复杂度至多为 $k$，而随机化查询复杂度为 $\tilde{\Omega}(n^{1-\frac{1}{2k}})$。这一有界错误量子与随机化查询复杂度的分离，根据 Aaronson 和 Ambainis（STOC 2015）以及 Bravyi、Gosset、Grier 和 Schaeffer（2021）的结果，是最优的。在我们工作之前，已知的最佳分离在多项式意义上更弱：对于任意 $\epsilon>0$，为 $O(1)$ 对比 $\Omega(n^{2/3-\epsilon})$（Tal, FOCS 2020）。作为另一个应用，我们得到了有界错误量子与随机化通信复杂度的本质最优分离：对于任意 $\epsilon>0$，为 $O(\log n)$ 对比 $\Omega(n^{1-\epsilon})$。之前的最佳分离在多项式意义上更弱：$O(\log n)$ 对比 $\Omega(n^{2/3-\epsilon})$（隐含于 Tal, FOCS 2020）。