Assuming P=BQP (ie we have polynomial time algorithms to solve all BQP problems) can we use it to prove that P=NP?


The argument is that since we have the Grover's algorithm which can solve NP complete problems with a quadratic speedup and since we have assumed that P=BQP, we can apply the Grovers algorithm repeatedly until it is reduced to a polynomial time problem.



What impact would P=BQP have on NP?
For what it’s worth, [Fortnow and Rogers](http://www.sciencedirect.com/science/article/pii/S0022000099916513) constructed an oracle relative to which P = BQP, but the polynomial hierarchy does not collapse (hence in particular P ≠ NP).


