QAC circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. QAC$^0$ circuits are QAC circuits of constant depth, and are quantum analogues of AC$^0$ circuits. We prove the following: $\bullet$ For all $d \ge 7$ and $\varepsilon>0$ there is a depth-$d$ QAC circuit of size $\exp(\mathrm{poly}(n^{1/d}) \log(n/\varepsilon))$ that approximates the $n$-qubit parity function to within error $\varepsilon$ on worst-case quantum inputs. Previously it was unknown whether QAC circuits of sublogarithmic depth could approximate parity regardless of size. $\bullet$ We introduce a class of "mostly classical" QAC circuits, including a major component of our circuit from the above upper bound, and prove a tight lower bound on the size of low-depth, mostly classical QAC circuits that approximate this component. $\bullet$ Arbitrary depth-$d$ QAC circuits require at least $\Omega(n/d)$ multi-qubit gates to achieve a $1/2 + \exp(-o(n/d))$ approximation of parity. When $d = \Theta(\log n)$ this nearly matches an easy $O(n)$ size upper bound for computing parity exactly. $\bullet$ QAC circuits with at most two layers of multi-qubit gates cannot achieve a $1/2 + \exp(-o(n))$ approximation of parity, even non-cleanly. Previously it was known only that such circuits could not cleanly compute parity exactly for sufficiently large $n$. The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which we name "nekomata" after an analogous cat y\=okai.

翻译：QAC 电路是量子电路,有1qubit的门和任意性的Toffoli的门。QAC $0 的电路是持续深度的QAC 电路,是AC $0 的量子电路的量子模拟值。 我们证明如下: 对于所有 $ ge 7 和 $ varepsilon > 0 的量子电路, $\ ball- 美元 QAC 电路的深度值为$( most- dd) QAC 电路的深度为$( mostly- more- more) (n/\ greal- dality) (n/ dipality) 美元 。对于低深度的量的量子电路程而言, 几乎是 QAC 平面的量值值值为$ $ 美元, 在正常的量子电路程中不能完全使用。