If P = NP were true, would quantum computers be useful?
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.