In this work, we prove the strongest known lower bounds for QAC$^0$, allowing polynomially many gates and ancillae. Our main results show that: (1) Depth-3 QAC$^0$ circuits cannot compute PARITY, and require $Ω(\exp(\sqrt{n}))$ gates to compute MAJORITY. (2) Depth-2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage, regardless of size. We develop new classical simulation techniques for QAC$^0$ to obtain our depth-3 bounds. In these results, we relax the output requirement of the quantum circuit to a single bit, making our depth $2$ approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC$^0$ circuits, which can compute PARITY exactly in depth $2$ (exp size). Our techniques further suggest that, for boolean total functions, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth $2$ QAC$^0$ circuits, regardless of size, cannot exactly synthesize an $n$-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth $2$ exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound). Finally, we argue that approximating PARITY in QAC0, with significantly better than 1/poly(n) advantage on average, is just as hard as computing it exactly. Thus, extending our techniques to higher depths would also rule out approximate circuits for PARITY and related problems

翻译：在本工作中，我们证明了目前已知的关于 QAC$^0$ 电路的最强下界，允许电路包含多项式数量的门和辅助量子位。我们的主要结果表明：（1）深度为 3 的 QAC$^0$ 电路无法计算 PARITY 函数，并且计算 MAJORITY 函数需要 $Ω(\exp(\sqrt{n}))$ 个门。（2）无论规模如何，深度为 2 的电路无法以不可忽略的优势逼近高影响力布尔函数（例如 PARITY）。我们为 QAC$^0$ 开发了新的经典模拟技术以获得深度 3 的下界。在这些结果中，我们将量子电路的输出要求放宽至单个比特，这使得我们的深度 2 逼近下界比 Rosenthal (2021) 之前的最佳下界更强。这也使我们能够与经典 AC$^0$ 电路进行自然的比较，后者可以在深度 2（指数规模）下精确计算 PARITY。我们的技术进一步表明，对于布尔全函数，常数深度量子电路不一定比其经典对应物提供更强的计算能力。我们的第三个结果表明，无论规模如何，深度为 2 的 QAC$^0$ 电路无法精确合成一个 $n$ 目标的猫又态（其合成与 PARITY 的计算直接相关）。这补充了 Rosenthal (2021) 关于逼近猫又态的深度 2 指数规模上界（该子电路被用于目前唯一已知的常数深度 PARITY 上界构造中）。最后，我们论证了在 QAC0 中以显著优于 1/poly(n) 的平均优势逼近 PARITY，其难度与精确计算该函数相当。因此，将我们的技术推广到更高深度也将排除用于 PARITY 及相关问题的近似电路。