A recent line of work has shown the unconditional advantage of constant-depth quantum computation, or $\mathsf{QNC^0}$, over $\mathsf{NC^0}$, $\mathsf{AC^0}$, and related models of classical computation. Problems exhibiting this advantage include search and sampling tasks related to the parity function, and it is natural to ask whether $\mathsf{QNC^0}$ can be used to help compute parity itself. We study $\mathsf{AC^0\circ QNC^0}$ -- a hybrid circuit model where $\mathsf{AC^0}$ operates on measurement outcomes of a $\mathsf{QNC^0}$ circuit, and conjecture $\mathsf{AC^0\circ QNC^0}$ cannot even achieve $\Omega(1)$ correlation with parity. As evidence for this conjecture, we prove: $\bullet$ When the $\mathsf{QNC^0}$ circuit is ancilla-free, this model achieves only negligible correlation with parity. $\bullet$ For the general (non-ancilla-free) case, we show via a connection to nonlocal games that the conjecture holds for any class of postprocessing functions that has approximate degree $o(n)$ and is closed under restrictions, even when the $\mathsf{QNC^0}$ circuit is given arbitrary quantum advice. By known results this confirms the conjecture for linear-size $\mathsf{AC^0}$ circuits. $\bullet$ Towards the a switching lemma for $\mathsf{AC^0\circ QNC^0}$, we study the effect of quantum preprocessing on the decision tree complexity of Boolean functions. We find that from this perspective, nonlocal channels are no better than randomness: a Boolean function $f$ precomposed with an $n$-party nonlocal channel is together equal to a randomized decision tree with worst-case depth at most $\mathrm{DT}_\mathrm{depth}[f]$. Our results suggest that while $\mathsf{QNC^0}$ is surprisingly powerful for search and sampling, that power is "locked away" in the global correlations of its output, inaccessible to simple classical computation for solving decision problems.

翻译：近期一系列工作展示了恒定深度量子计算（即$\mathsf{QNC^0}$）相对于$\mathsf{NC^0}$、$\mathsf{AC^0}$及相关经典计算模型具有无条件优势。体现这种优势的问题包括与奇偶函数相关的搜索和采样任务，自然引发了一个疑问：$\mathsf{QNC^0}$能否用于帮助计算奇偶函数本身？我们研究$\mathsf{AC^0\circ QNC^0}$——一种混合电路模型，其中$\mathsf{AC^0}$对$\mathsf{QNC^0}$电路的测量结果进行操作，并猜想$\mathsf{AC^0\circ QNC^0}$甚至无法实现与奇偶函数的$\Omega(1)$相关性。为支持这一猜想，我们证明： • 当$\mathsf{QNC^0}$电路无辅助比特时，该模型仅能实现与奇偶函数的可忽略相关性。 • 对于一般（含辅助比特）情况，通过与非局域游戏的关联表明：当后处理函数类具有近似度$o(n)$且在限制下封闭时，即使$\mathsf{QNC^0}$电路获得任意量子先验信息，该猜想仍成立。根据已知结果，这证实了线性规模$\mathsf{AC^0}$电路下的猜想。 • 为建立$\mathsf{AC^0\circ QNC^0}$的切换引理，我们研究量子预处理对布尔函数决策树复杂度的影响。我们发现，从该视角看，非局域信道并不比随机性更优：将布尔函数$f$与$n$方非局域信道预复合后，等价于最坏情况深度不超过$\mathrm{DT}_\mathrm{depth}[f]$的随机化决策树。我们的结果表明，尽管$\mathsf{QNC^0}$在搜索和采样任务中惊人地强大，但这种能力被“锁定”在其输出的全局相关性中，无法被简单经典计算用于解决判定问题。