If P = NP were true, would quantum computers be useful? Suppose that P = NP is true.  Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on the fact that P = NP is true?  How would you characterize the improvement in efficiency that would come about if a quantum computer could be built in a world where P = NP, as opposed to a world in which P != NP?

Here's a made-up example of about what I'm looking for:


  If P != NP, we see that complexity class ABC is equal to quantum complexity class XYZ...but if P = NP, class ABC collapses to related class UVW anyway.


(Motivation:  I am curious about this, and relatively new to quantum computing; please migrate this question if it is insufficiently advanced.)
The paper "BQP and the Polynomial Hierarchy" by Scott Aaronson directly addresses your question. If P=NP, then PH would collapse. If furthermore BQP were in PH, then no quantum speed-up would be possible in that case. On the other hand, Aaronson gives evidence for a problem with quantum speedup outside PH, thus such a speed-up would survive a collapse of PH.