QAC$^0$ is the class of constant-depth quantum circuits with polynomially many ancillary qubits, where Toffoli gates on arbitrarily many qubits are allowed. In this work, we show that the parity function cannot be computed in QAC$^0$, resolving a long-standing open problem in quantum circuit complexity more than twenty years old. As a result, this proves ${\rm QAC}^0 \subsetneqq {\rm QAC}_{\rm wf}^0$. We also show that any QAC circuit of depth $d$ that approximately computes parity on $n$ bits requires $2^{\widetilde{\Omega}(n^{1/d})}$ ancillary qubits, which is close to tight. This implies a similar lower bound on approximately preparing cat states using QAC circuits. Finally, we prove a quantum analog of the Linial-Mansour-Nisan theorem for QAC$^0$. This implies that, for any QAC$^0$ circuit $U$ with $a={\rm poly}(n)$ ancillary qubits, and for any $x\in\{0,1\}^n$, the correlation between $Q(x)$ and the parity function is bounded by ${1}/{2} + 2^{-\widetilde{\Omega}(n^{1/d})}$, where $Q(x)$ denotes the output of measuring the output qubit of $U|x,0^a\rangle$. All the above consequences rely on the following technical result. If $U$ is a QAC$^0$ circuit with $a={\rm poly}(n)$ ancillary qubits, then there is a distribution $\mathcal{D}$ of bounded polynomials of degree polylog$(n)$ such that with high probability, a random polynomial from $\mathcal{D}$ approximates the function $\langle x,0^a| U^\dag Z_{n+1} U |x,0^a\rangle$ for a large fraction of $x\in \{0,1\}^n$. This result is analogous to the Razborov-Smolensky result on the approximation of AC$^0$ circuits by random low-degree polynomials.

翻译：QAC$^0$是指允许任意多量子比特Toffoli门、具有多项式规模辅助量子比特的常数深度量子电路类。本工作中，我们证明了奇偶校验函数无法在QAC$^0$中计算，从而解决了量子电路复杂度领域一个悬置二十余年的公开问题。由此可证${\rm QAC}^0 \subsetneqq {\rm QAC}_{\rm wf}^0$。我们还证明：对于深度为$d$、在$n$比特上近似计算奇偶校验函数的任何QAC电路，需要$2^{\widetilde{\Omega}(n^{1/d})}$个辅助量子比特，该下界近乎紧确。这暗示了使用QAC电路近似制备猫态时存在类似下界。最后，我们证明了QAC$^0$的Linial-Mansour-Nisan定理的量子类比。该定理表明：对于具有$a={\rm poly}(n)$个辅助量子比特的任意QAC$^0$电路$U$及任意$x\in\{0,1\}^n$，$Q(x)$与奇偶校验函数的相关性以${1}/{2} + 2^{-\widetilde{\Omega}(n^{1/d})}$为界，其中$Q(x)$表示测量$U|x,0^a\rangle$输出量子比特的结果。以上所有结论均依赖于以下技术结果：若$U$是具有$a={\rm poly}(n)$个辅助量子比特的QAC$^0$电路，则存在一个由polylog$(n)$阶有界多项式构成的分布$\mathcal{D}$，使得以高概率从$\mathcal{D}$中随机选取的多项式能对大部分$x\in \{0,1\}^n$近似函数$\langle x,0^a| U^\dag Z_{n+1} U |x,0^a\rangle$。该结果类似于Razborov-Smolensky关于通过随机低阶多项式近似AC$^0$电路的经典结论。