AI 中文总结
该研究针对QAC⁰中对称布尔函数的扇出复杂度问题，证明计算对称布尔函数等价于实现其过渡半径对应的扇出，结合Paturi定理推导了PARITYₙ与QAC⁰中非对称函数的关联。
AI 中文摘要
QAC⁰能否计算PARITYₙ仍是开放问题。计算PARITYₙ等价于在QAC⁰归约下实现FANOUTₙ，这提出了更一般的问题：对任意对称布尔函数f:{0,1}ⁿ→{0,1}，在QAC⁰中计算f所需且充分的扇出大小是多少？我们证明答案恰好是过渡半径ρ(f)：在QAC⁰归约下，计算f与实现FANOUT_{ρ(f)}等价。特别地，若ρ(f)≥n^δ（δ>0为常数），则计算f是QAC⁰_f-完全的。结合Paturi定理，我们的刻画意味着：若PARITYₙ∉QAC⁰，则QAC⁰中近似度为n^{1/2+Ω(1)}的任意布尔函数必为非对称的。
英文摘要
Whether $\mathsf{QAC}^0$ can compute $\mathtt{PARITY}_n$ remains open. Computing $\mathtt{PARITY}_n$ is equivalent to implementing $\mathtt{FANOUT}_n$ under $\mathsf{QAC}^0$ reductions. This raises a more general question: for an arbitrary symmetric Boolean function $f:\{0,1\}^n\to\{0,1\}$, what fanout size is necessary and sufficient for computing $f$ in $\mathsf{QAC}^0$? We show that the answer is exactly the transition radius $ρ(f)$: computing $f$ and implementing $\mathtt{FANOUT}_{ρ(f)}$ are equivalent under $\mathsf{QAC}^0$ reductions. In particular, if $ρ(f)\ge n^δ$ for some constant $δ>0$, then computing $f$ is $\mathsf{QAC}^0_{\mathrm{f}}$-complete. Combined with Paturi's theorem, our characterization implies that if $\mathtt{PARITY}_n \notin \mathsf{QAC}^0$, then any Boolean function in $\mathsf{QAC}^0$ of approximate degree $n^{1/2+Ω(1)}$ must be nonsymmetric.
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