The circuit class $\mathsf{QAC}^0$ was introduced by Moore (1999) as a model for constant depth quantum circuits where the gate set includes many-qubit Toffoli gates. Proving lower bounds against such circuits is a longstanding challenge in quantum circuit complexity; in particular, showing that polynomial-size $\mathsf{QAC}^0$ cannot compute the parity function has remained an open question for over 20 years. In this work, we identify a notion of the \emph{Pauli spectrum} of $\mathsf{QAC}^0$ circuits, which can be viewed as the quantum analogue of the Fourier spectrum of classical $\mathsf{AC}^0$ circuits. We conjecture that the Pauli spectrum of $\mathsf{QAC}^0$ circuits satisfies \emph{low-degree concentration}, in analogy to the famous Linial, Nisan, Mansour theorem on the low-degree Fourier concentration of $\mathsf{AC}^0$ circuits. If true, this conjecture immediately implies that polynomial-size $\mathsf{QAC}^0$ circuits cannot compute parity. We prove this conjecture for the class of depth-$d$, polynomial-size $\mathsf{QAC}^0$ circuits with at most $n^{O(1/d)}$ auxiliary qubits. We obtain new circuit lower bounds and learning results as applications: this class of circuits cannot correctly compute -- the $n$-bit parity function on more than $(\frac{1}{2} + 2^{-\Omega(n^{1/d})})$-fraction of inputs, and -- the $n$-bit majority function on more than $(1 - 1/\mathrm{poly}(n))$-fraction of inputs. \end{itemize} Additionally we show that this class of $\mathsf{QAC}^0$ circuits with limited auxiliary qubits can be learned with quasipolynomial sample complexity, giving the first learning result for $\mathsf{QAC}^0$ circuits. More broadly, our results add evidence that ``Pauli-analytic'' techniques can be a powerful tool in studying quantum circuits.

翻译：电路类$\mathsf{QAC}^0$由Moore（1999）引入，作为恒定深度量子电路的模型，其中门集合包含多量子比特Toffoli门。证明此类电路的下界是量子电路复杂性领域的一个长期挑战；特别是，证明多项式规模的$\mathsf{QAC}^0$无法计算奇偶函数，二十多年来一直是一个未解难题。在这项工作中，我们定义了$\mathsf{QAC}^0$电路的\emph{泡利谱}概念，这可以视为经典$\mathsf{AC}^0$电路傅里叶谱的量子类比。我们猜想，$\mathsf{QAC}^0$电路的泡利谱满足\emph{低度集中性》，类似于经典的Linial、Nisan、Mansour关于$\mathsf{AC}^0$电路低度傅里叶集中性的著名定理。若此猜想成立，则立即意味着多项式规模的$\mathsf{QAC}^0$电路无法计算奇偶函数。我们针对深度为$d$、多项式规模且至多使用$n^{O(1/d)}$个辅助量子比特的$\mathsf{QAC}^0$电路证明了该猜想。作为应用，我们获得了新的电路下界和学习结果：此类电路无法正确计算——$n$比特奇偶函数在超过$(\frac{1}{2} + 2^{-\Omega(n^{1/d})})$的输入上，以及——$n$比特多数函数在超过$(1 - 1/\mathrm{poly}(n))$的输入上。此外，我们证明了这类有限辅助量子比特的$\mathsf{QAC}^0$电路可以拟多项式样本复杂度学习，给出了$\mathsf{QAC}^0$电路的第一个学习结果。更广泛地说，我们的结果增添了证据，表明“泡利分析”技术可以成为研究量子电路的强大工具。