Fanout Complexity of Symmetric Boolean Functions in $\mathsf{QAC}^0$

Abstract: Whether $\mathsf{QAC}<sup>0$ can compute $\mathtt{PARITY}<em>n$ remains open. Computing $\mathtt{PARITY}_n$ is equivalent to implementing $\mathtt{FANOUT}_n$ under $\mathsf{QAC}<sup>0$ reductions. This raises a more general question: for an arbitrary symmetric Boolean function $f:{0,1}<sup>n\to{0,1}$, what fanout size is necessary and sufficient for computing $f$ in $\mathsf{QAC}<sup>0$? We show that the answer is exactly the transition radius $ρ(f)$: computing $f$ and implementing $\mathtt{FANOUT}</em>{ρ(f)}$ are equivalent under $\mathsf{QAC}<sup>0$ reductions. In particular, if $ρ(f)\ge n<sup>δ$ for some constant $δ&gt;0$, then computing $f$ is $\mathsf{QAC}<sup>0_{\mathrm{f}}$-complete. Combined with Paturi's theorem, our characterization implies that if $\mathtt{PARITY}_n \notin \mathsf{QAC}<sup>0$, then any Boolean function in $\mathsf{QAC}<sup>0$ of approximate degree $n<sup>{1/2+Ω(1)}$ must be nonsymmetric.