Actually Penrose isn't so much interested in qualia as he is in things mathematicians can do, but which Turing machines cannot (from http://en.wikipedia.org/wiki/Quantum_mind - "Penrose is not interested in explaining phenomenal consciousness, qualia, generally regarded as the most mysterious feature of consciousness, but instead focuses mainly on the cognitive powers of mathematicians").

If I have closed timelike curves, an oracle for the halting problem, or the ability to perform arbitrary computations on real numbers in my brain, I would certainly like to know how to use them.