I don't think that's really true. The difference between a quantum computer and a classical computer is that a classical computer can represent N states given N

I don't think that's really true. The difference between a quantum computer and a classical computer is that a classical computer can represent N states given N bits, whereas a quantum computer can represent 2^N states given N q-bits. However, this has a major caveat: you can never know the state of the system. You only get to measure one value at a time and you can never re-create or copy a previously obtained state. Still, some clever people have figured out how to exploit this to achieve polynomial speedup on problems where certain properties of the problem are exploited to get around the limitation of measurement in a quantum system.