if you could build one, actually lets you solve more problems efficiently than you could with a classical computer; this is the question of whether P = BQP.

This isn't quite what this question asks. We know that some things are faster with a quantum computer than what can be one in the best case classical situation. Grover's algorithm is one example. The question of whether P=BQP is one formulation of that for yes and no questions. But even this isn't quite accurate. For example, it could be that quantum computers can do some extremely high degree polynomial speed-up even while P=BQP. But this is essentially a minor nitpick.