I agree that the superposition with complex probability amplitudes (fundamental quantum randomness) is different from just statistical randomness of trying every possibility. I fact, this is one of the notions which is being used to verify whether the D-Wave computer (as a black box) is "truly quantum".
1. I don't quite see what periodicity has to do with the speedup. Could you elaborate on that?
2. "...if they did that you'd be able to solve NP problems in constant time." Again, please explain.
In QM, the answer you calculate is actually the weighted sum over all possibilities... so the way I see it, it does cover every possible path. That answer might not be useful in telling you which is the shortest single path, but if you were an electron traversing a graph and wanted to optimize your travel, you'd do exactly what the quantum computer says and traverse all paths! The classical notion of "best path" might be trickier to deduce from a quantum computer. Maybe I'm addressing something tangential to what you're saying.
1. I don't quite see what periodicity has to do with the speedup. Could you elaborate on that?
2. "...if they did that you'd be able to solve NP problems in constant time." Again, please explain. In QM, the answer you calculate is actually the weighted sum over all possibilities... so the way I see it, it does cover every possible path. That answer might not be useful in telling you which is the shortest single path, but if you were an electron traversing a graph and wanted to optimize your travel, you'd do exactly what the quantum computer says and traverse all paths! The classical notion of "best path" might be trickier to deduce from a quantum computer. Maybe I'm addressing something tangential to what you're saying.