I think that it is at least BQP. Dorit Aharonov and Ayal Green showed that PostBQP is contained in IP[BPP, PostBQP] (interactive protocol with a BPP verifier and a PostBQP verifier.)
Before someone points it out, I know that if I assume LWE (or technically speaking, the existence of an extended trapdoor claw-free family) then this follows from Urmila Mahadev’s breakthrough result from earlier this year but I don’t want to assume anything.
My current approach is to show that an additive approximation to the Jones polynomial is contained in this class. But I don’t know how to make it work. (I only spent half a day on it so maybe it is obvious and I just missed it.)
We know how to do this for easier problems; for instance, François Le Gall, Tomoyuki Morimae, Harumichi Nishimura, and Yuki Takeuchi showed that computing orders of solvable groups (which John Watrous put in BQP) is in IP[BPP, BQP].
While trying to write up a different result (which is what I should be doing), I stumbled upon Lance Fortnow’s thesis and it is awesome! (I should prolly get back to writing…)