From what I've read on Scott Aaronson's blogs, NP-complete is also wrong. The general suspicion seems to be that polynomial quantum algorithms (BQP) are separate from NP - they are suspected to be neither a subset nor a superset of NP.
Some problems in BQP are suspected to be in NP (integer factorization, for which the best known classical algorithm is sub-exponential, but we have a polynomial time quantum algorithm), but there is no known NP-complete problem for which a quantum algorithm is known, or even suspected to exist.
> If we interpret the space of 2n possible assignments to a Boolean formula φ as a “database,” and the satisfying assignments of φ as “marked items,” then Bennett et al.’s result says that any quantum algorithm needs at least ∼n/2 steps to find a satisfying assignment of φ with high
probability, unless the algorithm exploits the structure of φ in a nontrivial way. In other words, there is no “brute-force” quantum algorithm to solve NP-complete problems in polynomial time, just as there is no brute-force classical algorithm.
Some problems in BQP are suspected to be in NP (integer factorization, for which the best known classical algorithm is sub-exponential, but we have a polynomial time quantum algorithm), but there is no known NP-complete problem for which a quantum algorithm is known, or even suspected to exist.
Edit - some links:
[0] https://www.scottaaronson.com/papers/npcomplete.pdf - chapter 4
> If we interpret the space of 2n possible assignments to a Boolean formula φ as a “database,” and the satisfying assignments of φ as “marked items,” then Bennett et al.’s result says that any quantum algorithm needs at least ∼n/2 steps to find a satisfying assignment of φ with high probability, unless the algorithm exploits the structure of φ in a nontrivial way. In other words, there is no “brute-force” quantum algorithm to solve NP-complete problems in polynomial time, just as there is no brute-force classical algorithm.
[1] https://youtu.be/0jrybODBUpA?t=30m28s "P versus NP"