Parity vs. $\mathsf{AC^0}$ with simple quantum preprocessing

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.