We prove that for every decision tree, the absolute values of the Fourier coefficients of 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 any positive integer $k,$ a partial Boolean function on $n$ bits that has bounded-error quantum query complexity at most $\lceil k/2\rceil$ and randomized query complexity $\tilde{\Omega}(n^{1-1/k}).$ This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015). Prior to our work, the best known separation was polynomially weaker: $O(1)$ versus $n^{2/3-\epsilon}$ for any $\epsilon>0$ (Tal, FOCS 2020).

中文翻译：

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随机和量子查询复杂性的最佳分离

我们证明对于每个决策树，给定阶数 $\ell\geq1$ 的傅立叶系数的绝对值总和至多为 $c^{\ell}\sqrt{\binom{d}{\ell}(1+ \log n)^{\ell-1}},$ 其中 $n$ 是变量的数量，$d$ 是树的深度，$c>0$ 是一个绝对常数。由于 Tal（arxiv 2019；FOCS 2020），这个界限本质上是紧密的，并且解决了一个猜想。我们工作之前的边界随着 $\ell,$ 在 $\ell=\sqrt{d} 处变得微不足道而迅速下降。作为一个应用程序，我们获得了对于任何正整数 $k,$ $ 上的部分布尔函数n$ 位，有界误差量子查询复杂度至多 $\lceil k/2\rceil$ 和随机查询复杂度 $\tilde{\Omega}(n^{1-1/k}).$ 这种有界分离- 根据 Aaronson 和 Ambainis (STOC 2015) 的结果，错误量子与随机查询复杂性是最好的。