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 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 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 tasks, 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（$\mathsf{QNC^0}$）相对于$\mathsf{NC^0}$、$\mathsf{AC^0}$及相关经典计算模型的无条件优势。展现这一优势的问题包括与奇偶函数相关的搜索和采样任务，而自然的问题是$\mathsf{QNC^0}$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$\mathsf{AC^0\circ QNC^0}$无法达到与奇偶函数$\Omega(1)$的相关性。作为这一猜想的证据，我们证明： - 当$\mathsf{QNC^0}$电路无辅助量子比特时，该模型与奇偶函数的相关性可忽略。 - 对于一般（含辅助量子比特）情形，通过与非局域博弈的联系，我们证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$\mathsf{QNC^0}$电路被赋予任意量子咨询，该猜想依然成立。根据已知结果，这证实了线性规模$\mathsf{AC^0}$电路的猜想。 - 针对$\mathsf{AC^0\circ QNC^0}$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。从这一视角发现，非局域信道并不优于随机性：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至多为$\mathrm{DT}_\mathrm{depth}[f]$的随机决策树。我们的结果表明，尽管$\mathsf{QNC^0}$在搜索和采样任务中出奇强大，但这种能力被"锁在"其输出的全局关联中，无法被简单经典计算用于求解决策问题。