Establish whether \(\mathsf{QAC}^0\) contains Boolean functions of large approximate degree

Establish whether the quantum constant-depth circuit class \(\mathsf{QAC}^0\) contains Boolean function families whose approximate degree grows substantially with the input length, in particular of order \(n^{1/2+\Omega(1)}\) or larger.

Background

Approximate degree is the minimum degree of a real polynomial that approximates a Boolean function pointwise to error at most $1/3$. The paper invokes Paturi’s characterization to relate the approximate degree of a nonconstant symmetric Boolean function to its transition radius: $\widetilde{\deg}(f)=\Theta(\sqrt{n\rho(f)})$.

Using this relation, the paper proves that any symmetric function with approximate degree $\Omega(n^{1/2+\eta})$ is $\mathsf{QAC}^0_{\mathrm f}$-complete under the stated worst-case computation criterion. The unresolved issue is broader: whether $\mathsf{QAC}^0$ contains any Boolean functions of large approximate degree at all, including potentially nonsymmetric functions.