$\mathsf{QAC}^0$ is the class of constant-depth polynomial-size quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates. It is arguably the smallest natural class of constant-depth quantum computation which has not been shown useful for computing any non-trivial Boolean function. Despite this, many attempts to port classical $\mathsf{AC}^0$ lower bounds to $\mathsf{QAC}^0$ have failed. We give one possible explanation of this: $\mathsf{QAC}^0$ circuits are significantly more powerful than their classical counterparts. We show the unconditional separation $\mathsf{QAC}^0\not\subset\mathsf{AC}^0[p]$ for decision problems, which also resolves for the first time whether $\mathsf{AC}^0$ could be more powerful than $\mathsf{QAC}^0$. Moreover, we prove that $\mathsf{QAC}^0$ circuits can compute a wide range of Boolean functions if given multiple copies of the input: $\mathsf{TC}^0 \subseteq \mathsf{QAC}^0 \circ \mathsf{NC}^0$. Along the way, we introduce an amplitude amplification technique that makes several approximate constant-depth constructions exact.

翻译：$\mathsf{QAC}^0$是由任意单量子比特门和广义托佛利门构成的常数深度多项式规模量子电路类。它可以说是尚未被证明能用于计算任何非平凡布尔函数的最小自然常数深度量子计算类。尽管如此，许多将经典$\mathsf{AC}^0$下界移植到$\mathsf{QAC}^0$的尝试均告失败。我们为此提供了一种可能的解释：$\mathsf{QAC}^0$电路比其经典对应物显著更强大。我们证明了决策问题的无条件分离$\mathsf{QAC}^0\not\subset\mathsf{AC}^0[p]$，这同时也首次解决了$\mathsf{AC}^0$是否可能比$\mathsf{QAC}^0$更强大的问题。此外，我们证明若给定输入的多个副本，$\mathsf{QAC}^0$电路能够计算广泛的布尔函数：$\mathsf{TC}^0 \subseteq \mathsf{QAC}^0 \circ \mathsf{NC}^0$。在此过程中，我们引入了一种振幅放大技术，使得若干近似常数深度构造变得精确。