Evidence that there is some problem in BQP distinct from BPP? Are there any evidences (1 physics, 2 mathematics AND 3 computer science) that particular problems such as integer factorization, discrete logarithm are in BQP but not in BPP? There do not seem to be any classical collapse results if these problems are in BPP. Is there reason to think BQP is certainly a class different from BPP?
Scott Aaronson has been addressing this topic: http://arxiv.org/abs/0910.4698

Related hardness results:


Boson sampling: http://arxiv.org/abs/1011.3245
Commuting circuits: http://arxiv.org/abs/1005.1407
DQC1 model: http://arxiv.org/abs/1509.07276
Extended Clifford circuits: http://arxiv.org/abs/1305.6190