There is something very strange and absurd for me about the power of a quantum computer. Let me briefly states the following facts:
Fact 1: theoretical computer scientists believe (very likely to happen) that quantum computer cannot solve NP-complete problems.
Fact 2: In BQP, there are problems that is harder than NP, such as Boson sampling distribution which is known to be $\#P$$-hard$.
Fact 3: If Boson sampling distribution (and other problems such as Quantum Sampling) belong to $PH$, then the polynomial time hierarchy collapsed. So, it is unlikely that these problems belongs to $PH$.
Now, by these facts it is said that BQP is greater class than $PH$. I.e. since all NP problems lies in PH, then BQP contains all NP problems including the NP-complete problems. But, at the same time, theoretical computer scientists (such as Scott Aaronson) conjectured that quantum computer cannot solve NP-complete problems efficiently, i.e. they are trying to say the following: BQP should be smaller than class PH.
Can someone explain to me whether I'm wrong about something or not.