People say it's a misconception that quantum computers can evaluate a function on all its possible inputs in parallel. But actually a lot of quantum algorithms do begin by applying a function to a superposition of all possible inputs. It's just that after that point you need to do some difficult linear algebra and you can't always extract the information you want.
In fact, you can define a lot of important complexity classes in this way. The set of problems solvable in polynomial time with an oracle that evaluates a given circuit on all its possible inputs and tells you …
P: … nothing.
BPP: … a randomly chosen output.
PP: … a majority output.
NP: … if any of the outputs is nonzero.
co-NP: … if all of the outputs are nonzero.
PSPACE: … a fixed point.