@narip That achieves the same goal, but I would like to know if the method I proposed is possible.

@MarkS Thanks for the response; your comment answers my question. The title was just asking if a non-promise problem is known to be BQP-complete, and if it did exist, why it isn't easy to show it is in NP. Yeah, it isn't always trivial to determine if a problem is in NP, I guess that part of my question isn't completely true.

Oh, I guess I have to study more about this.

@glS That could certainly be true. But I don't think its correct because the product-test (the algorithm they use to prove that this problem is in BQP) doesn't have this $\alpha$ and $\beta$ closeness/farness specification. It only requires an error $\epsilon$. Maybe this way introduced for the $1$-LOCC version?

@glS Oh, that looks true. But how would we use a quantum-classical channel on a matrix that isn't a density matrix?

@glS Oh. I saw that quote, but I don’t see how that can be faster than a SWAP test. It seems like they are suggesting something similar to tomography.