2-Fold Forrelation is in QAC${}^0$

Abstract

We show that 2-fold Forrelation with inverse-polylogarithmic promise gap can be solved, with bounded error, by polynomial-size QAC${}^0$ circuits. Unlike the standard oracle-based Forrelation algorithm, our circuits receive the input explicitly, in the same form as the AC${}^0$ circuits against which Forrelation is known to be hard. At constant gap, this yields a natural promise-problem separation between QAC${}^0$ and AC${}^0$.