QAC0 Can Prepare Every Logarithmic-Qubit State

Abstract: $\mathsf{QAC}<sup>0$ is the class of constant-depth $\mathrm{poly}(n)$-ancilla circuits obtained by extending $\mathsf{QNC}<sup>0$, the class of local circuits, to include nonlocal interactions via arbitrary width Toffoli gates. It is believed to be weaker than its counterpart, $\mathsf{QNC}<sup>0_f$, obtained by including arbitrary-width FANOUT gates instead ($$\mathsf{QAC}⁰ \subseteq \mathsf{QNC}<sup>0_f$$ [Moo99]). In this note, we show that every $O(\log n)$-qubit state can be exactly and cleanly prepared by a $\mathrm{poly}(n)$-ancilla $\mathsf{QAC}<sup>0$ circuit. Previous known $\mathrm{poly}(n)$-ancilla circuits for arbitrary such states are only known via additional access to either FANOUT or QRAM (indexing) gates [Ros21b, GGJ26b], neither of which are known to be in $\mathsf{QAC}<sup>0$. Equivalently, prior constructions of arbitrary $n$-qubit states in $\mathsf{QAC}<sup>0$ require doubly exponential size and we obtain an exponential factor improvement.