Improved Lower Bounds for QAC0

Malvika Raj Joshi; Avishay Tal; Francisca Vasconcelos; John Wright

Paper

Improved Lower Bounds for QAC0

Abstract

In this work, we establish the strongest known lower bounds against QAC

, while allowing its full power of polynomially many ancillae and gates. Our two main results show that: (1) Depth 3 QAC

circuits cannot compute PARITY regardless of size, and require at least

many gates to compute MAJORITY. (2) Depth 2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage in depth

, regardless of size. We present new techniques for simulating certain QAC

circuits classically in AC

to obtain our depth

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

approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC

circuits, which can compute PARITY exactly in depth

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

QAC

circuits, regardless of size, cannot exactly synthesize an

-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth

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).