The computational power of the human brain

You can use a Turing machine to (inefficiently) simulate a quantum computer just fine.

The page you linked to also seems to think that Gödel's incompleteness theorem says that human brains can do thinks that computers can't. That conclusion isn't warranted:

Yes, there are specific statements that specific formal systems can neither prove nor disprove. And for every formal system you can find such systems.

However, there are also plenty of statements that humans can neither prove nor disprove. Human also make plenty of mistakes. (You can also build a probabilistic formal system that's allowed to make mistakes.)

> This leads to a fundamental point about any attempt to understand the brain. It is a particularly acute case of the problems encountered in understanding anything. The nature of the attention one brings to bear on anything alters what one finds; what we aim to understand changes its nature with the context in which it lies; and we can only ever understand anything as a something.

> There is no way round these problems—if they are problems. To attempt to detach oneself entirely is just to bring a special kind of attention to bear which will have important consequences for what we find. Similarly we cannot see something without there being a context, even if the context appears to be that of ‘no context’, a thing ripped free of its moorings in the lived world. That is just a special, highly value-laden kind of context in itself, and it certainly alters what we find, too. Nor can we say that we do not see things as anything at all—that we just see them, full stop. There is always a model by which we are understanding, an exemplar with which we are comparing, what we see, and where it is not identified it usually means that we have tacitly adopted the model of the machine.

>

> Such considerations apply to the attempt to understand anything at all. But when we come to look at what we refer to as brain functions, there is a problem of a wholly different order. We are not ‘just’ looking at things in the world—a lump of rock, or even a person—but the processes whereby the world itself, together with the rock or the person, might be brought into being for us at all, the very foundations of the fact of our experience, including any idea we might have about the nature of the world, and the nature of the brain, and even the idea that this is so. If it is true that attention changes the nature of what we find, how do we decide the most appropriate attention for that? One that tries to ignore the inwardness of experience? What possible context is there in which to place the foundations of experience of all contexts whatever? And what kind of a thing are we to see it ‘as’? The answer is far from obvious, but in the absence of an attempt to address the question we do not give no answer. We answer with the model we understand—the only kind of thing we can ever fully understand, for the simple reason that we made it: the machine.

Meanwhile we have abundant evidence that continuous variables such as the precise timing of individual spikes have physiological effects.

But then of course, any analog process can be simulated by a sufficiently high fidelity digital one (i.e. digital music encoding). So all you've really got to is "it might take a considerably more powerful computer to fully reproduce the analog effects of human consciousness".