There are two forms of computing currently thought to be possible in our universe. One is the classical, deterministic computing that we all know and love. Many people think the human brain is a kind of (very large and complicated) classical computer. However, it is still unknown whether (and if so, how) a classical computer can give rise to consciousness and subjective experience.
The second form of computing is quantum computing, where you essentially run a pile of classical computers in superposition and allow their outputs to interfere in order to obtain the result. Anything quantum computers can do can also be done by classical computers (albeit much more slowly). The human brain might be a quantum computer, but (unless there's something about quantum computing that we don't yet understand) that still doesn't solve the problem of consciousness.
A third form of computing is possible if you have a time machine. I've speculated before that the human brain could be a time travelling computer. These computers are faster still than quantum computers, but still can't compute anything that can't in principle (given long enough) be computed by a classical computer, so this still doesn't solve the consciousness problem.
Could it be that by accident of evolution the human brain has tapped into a form of computing that is qualitatively different from classical computing, much as birds and bees have tapped into a qualitatively different method of flying (flapping) than the method use in our aeroplanes? While this smells of dualism, I think it's a possibility that can't be fully discounted without a complete theory of physics.
One such qualitatively different form of computing is the infinity machine. This can verify true things in finite time even if there is no finite proof that those things are true. Thus it can find completely new truths that are not provable by conventional mathematics.
It seems rather unlikely that the infinity machine is possible in our universe (quantum mechanics puts an absolute limit on clock speed) but there could be other forms of computation that we've just never thought of.
Penrose's Orchestrated Objective Reduction theory is one such possibility.