Quantum computing actually isn't super-Turing, it "just" computes some things faster. (Strictly speaking it's somewhere between a standard Turing machine and a nondeterministic Turing machine in speed, and the first can emulate the second.)

If we're nitpicking: quantum computing algorithms could (if implemented) compute certain things faster than the best classical algorithms we know. We don't know any quantum algorithms that are provably faster than all possible classical algorithms.

Well yeah, we haven't even proved that P != NP yet.