I am aware that it is generally believed that P = BPP, but BQP != P (since factoring is in BQP, and factoring seems hard.)
For BPP, we have the hardness vs randomness result: which states that circuit lower bounds for EXP imply derandomization results for BPP.
Even though I do not expect BQP to be simulatable in sub-exponential time, I am curious if there are any implication results along the lines of "If pigs can whistle, than donkeys can fly."
Formally, I'm asking if there exists provable statements of the form:
If [XYZ] then BQP can be simulated in sub-exponential time.
Ideally, [XYZ] would be about the hardness of certain problems against certain complexity classes. However, any XYZ not of the form "BQP can be simulated in sub-exponential time" would be interesting.