Penrose believes that the brain may use certain quantum phenomena which enable it to solve non-Turing-computable problems. These include mathematical intuition and musical creativity, because they require a kind of holistic global perception that no algorithm can provide. He describes these views in detail in The Emperor's New Mind (KH:P something).
There is a demonstration that the brain can compute what's not Turing-computable (I think) on page 64 of his book.