Quantum lower bound for recursive Fourier sampling

We revisit the oft-neglected `recursive Fourier sampling' (RFS) problem, introduced by Bernstein and Vazirani to prove an oracle separation between BPP and BQP. We show that the known quantum algorithm for RFS is essentially optimal, despite its seemingly wasteful need to uncompute information. This implies that, to place BQP outside of PH relative to an oracle, one would need to go outside the RFS framework. Our proof argues that, given any variant of RFS, either the adversary method of Ambainis yields a good quantum lower bound, or else there is an efficient classical algorithm. This technique may be of independent interest.