TITLE: What impact would P!=NP  have on the characterization of BQP?
QUESTION [11 upvotes]: Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever solving NP-Complete problems in Quantum Polynomial time?
According to Wikipedia, quantum complexity classes BQP and QMA are the bounded-error quantum analogues of P and NP. Is it likely that a proof that $P\ne NP$ can be adapted to the quantum setting to show that $BQP \ne QMA?$

REPLY [2 votes]: To your second question. It is unlikely that the current under-review proof of P != NP will allow you to seperate BQP and QMA (or BQP and P, or BQP and NP, or even BPP and NP...). Deolalikar's proof uses descriptive complexity, in particular it uses a correspondence between statements expressible in certain logics and the complexity classes P and NP. As far as I know there is no such nice one-to-one correspondences between BPP, BQP, or QMA and nice logics.