I'm not a physicist, but my understanding is that the debate about whether or not "quantum is deterministic" centers around what your interpretation is of the wave function. I think most physicists, who Penrose would probably criticize, believe the "Copenhagen Interpretation," which says that the wave function of a system is literally the function that, when you take the inner product with its conjugate, gives you the probability distribution of the state of the system, and hence, the state is non-deterministic.

The uncertainty principle arises from quantum. Specifically, it's a statement about the commutativity of operators that act on the wave function, which, going by the Copenhagen interpretation of physics, means it's a consequence of the non-determinism of the universe. I think this is a philosophically satisfying explanation for why getting the state of the universe is impossible.