Improved Lower Bounds for QAC0

In this work, we establish the strongest known lower bounds against QAC$^0$, while allowing its full power of polynomially many ancillae and gates. Our two main results show that: (1) Depth 3 QAC$^0$ circuits cannot compute PARITY regardless of size, and require at least $Ω(\exp(\sqrt{n}))$ many gates to compute MAJORITY. (2) Depth 2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage in depth $2$, regardless of size. We present new techniques for simulating certain QAC$^0$ circuits classically in AC$^0$ to obtain our depth $3$ lower bounds. In these results, we relax the output requirement of the quantum circuit to a single bit (i.e., no restrictions on input preservation/reversible computation), making our depth $2$ approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC$^0$ circuits, which can compute PARITY exactly in depth $2$ using exponential size. Our proof techniques further suggest that, for inherently classical decision problems, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth $2$ QAC$^0$ circuits, regardless of size, cannot exactly synthesize an $n$-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth $2$ exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound).