Evidence that there is some problem in BQP distinct 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