First, a clarification: Turing machines are not powerful in any meaningful sense of the word. Instead, Turing machines are a simplified abstraction of the way in which calculation is performed. It doesn't make any sense to say that a Turing machine is the most powerful computer that can be made; the correct phrase is that any (current) computer, no matter how powerful, is at heart a Turing machine.

Second, only symbolic processing devices with finite internal states and strict sequencing rules qualify as Turing machines. Every conventional (current) computer can be modeled that way, but quantum computers and analog computers — and arguably other types of computers not yet invented — cannot be modeled as Turing machines. I'm not suggesting these computers are more 'powerful' than Turing-machine computers; they're merely different.

It's also worth pointing out that Turing machines were intended to reflect the human method of 'doing calculations', not of 'solving problems', and we shouldn't fall into the ideological trap of believing that all problems are matters of mere calculation. But's maybe that's just me being philosophically peevish…

But let me get to the point. The way things are framed here, this question seems to presume that the universe as a while is already something like a vast quantum computer. And this forces us to consider a fundamental difference. In Turing devices the machine is an external device that works on symbols it is completely divorced from. It's all a bit Cartesian: a Turing device doesn't know anything about the symbols it processes, and the symbols themselves never enter into the device, the device merely carries the tape that contains the symbols, moving it this way and that, while 'information' is worked out entirely on a symbolic level. Quantum devices, by contrast, are not separate from or external to the information they work on; quantum devices are the information they work on. When we look at one of the current efforts at quantum computing we will see a lot of machinery, obviously, but this machinery has nothing to do with the quantum computation itself. These machines are there to maintain the quantum elements in a particular state: cooling them to low temperatures, excluding vibrational noise and radiation, providing interfaces, probably a few flashing lights to make it look cool… The actual 'computation' is contained entirely within the quantum processor itself, as transformations of its quantum state.

Obviously, a universe that is itself a quantum processor could allow the creation of other quantum computers within itself (with the caveat that these localized quantum computers need to be isolated from confounding interactions with the universal processor). Whether this is 'powerful' is a separate issue. Much like AI, quantum devices are a neat idea that might have important specialized uses, but that may not be all that productive for general consumption. Most of the things we use smart phones for are matters of symbolic processing. Talking to people, sending texts, playing games, taking pictures, translating, looking up information: these are all symbolic transformations of one sort or another, and a Turing-type machine is (in some ways) ideally suited for that kind of symbolic transformation work. I can imagine quantum computers being useful for certain scientific tasks: things where fundamental principles can be 'coded in', and complex results can be achieved in time-frames that Turing devices can't match. But I can't imagine what use a hand-held quantum device could be put to. And almost certainly any hand-held quantum device would need to be paired with a sophisticated Turing device that would translate the user's symbolic commands into rules for the quantum device to execute, and then retranslate the output into humanly accessible symbols. If we have to have that Truing device anyway, what added value does the quantum element bring? Time will tell, I suppose, but right now a common-use quantum device seems a bit like strapping a jet engine on a Kia Sorento: cool as hell if one could do it, but not useful for any of the normal purposes of an SUV.

if we were to think of this universe itself as some machine, then it carries out far more powerful computations than a Turing machine. This is because the values of the fields in a region could correspond to non computable, or even non describable functions

If, just for the sake of idle speculation, we assume that those "values" are non-describable, then presumably this means: non-describable by us, by any means, ever. So, we cannot describe them. Thus, we cannot know about them. Thus, assuming that whatever happens in the universe is a kind of computation, but that no conceivable computation could carry it out, then we cannot simulate it, model it, conceive of it, understand it. In other words: we could never know this.

Quantum computers are also turing machines; they are not more powerful in the sense that they would enable us to compute thing that are essentially uncomputable by any turing machine (e.g. the halting problem); they just allow faster, more efficient computations. It's possible to imagine devices that are more powerful (Turing himself imagined an "oracle" that could answer certain uncomputable questions), but those devices can not be physically realized (if we believe the Church-Turing thesis).

The upshot of this is that, if the Church-Turing thesis is true, then the universe can also not be (operating as) a kind of device that is essentially more powerful than a turing machine (unless the universe itself is actually infinite or contains actual infinities).

Independent of that: an emergent computation can never be more powerful than the underlying system: it cannot compute something that the underlying system would not be able to compute. Even though some problems or tasks may only become "visible" at a global, emergent macro-level (for instance: determining parity by a 1-dimensional black-white cellular automaton; 'parity' is itself a global property). Emergence means that the same process can be seen in different ways, either on a micro-level or on a macro-level; the two views are still views of the very same underlying process however. (New information can never as it were spontaneously be created.)

I think this is a good question, but I also think it's very difficult to answer, and doing so would require some sophisticated mathematics.

Part of that would be defining exactly what's meant by the various components of the question. For example:

• What kind of thing are the laws of physics in this universe? Are they just arbitrary partial differential equations on a 4 dimensional manifold, or do they obey the constraints of Hamiltonian dynamics (or some other constraints)? Does the manifold change dynamically as in general relativity (and if so, is that considered an essential part of the question or can it be ignored?)? What are the boundary conditions - are they arbitrary or do you impose some constraints on them? Either way, do you demand that these partial differential equations have a unique solution across all space-time for any permissible boundary conditions, or do you allow the possibility of multiple solutions, partial solutions, singularities etc.?
• What is a computer in this setting exactly? How do you give it an input and get its output? Are the inputs and outputs continuous or discrete, and if the latter, how do you encode discrete signals in such a continuous universe?
• Are computers meant to be "macroscopic" systems in this world, as they are in ours? If so, how can a theory of statistical mechanics be developed for this universe, such that this claim can be made precise?

These points are not meant as a criticism of the question - I'm not suggesting you should try to clarify your question by answering them. (Although if you do have precise enough answers to them, you could try asking this to mathematicians.) Rather, I'm trying to illustrate that questions like this can be very subtle, and making the question precise enough to be answered can be a research programme in itself.

On the other hand, a Google search for "computability in continuous systems" resulted in several papers about continuous models of computation. These are unlikely to exactly be answers to your question, but they are quite likely to be useful if you want to get closer to it. People (including the great Claude Shannon) have already done some of the work of making this kind of problem more precise, and learning about that will probably be very helpful to you.

No, because you haven't understood the difference between continuous and discrete processing

Turing Machines are a model of discrete sequential processing. That is, operations are carried out one step at a time, with some delay between operations. It turns out that all discrete sequential processing can be translated into Turing Machine operations.

Turing Machines fundamentally do not represent continuous operations though. We get a pretty good approximation by reducing the sample rate, but it's still not really the same.

Back in the day, we did do analogue computing, which is truly continuous. Look up how gunsights and predictors worked in WW2, for example. They have mostly been replaced with digital technology as the ability to model with faster sample rates has become possible. This is not because digital allows us to do anything different, but simply because analogue computing is hard to design, hard to build, and hard to change if you need a slightly different algorithm.

You seem to be mixing up lots of things, mainly configuration space and "computing power". In our computers, small files can be stored on large hard disks but not the other way around, and similarly, finitely many numbers can be represented be countably infinitely many which in turn can be represented by uncountably infinitely many but not the other way around. This says nothing about speed/efficiency/whatnot of compute.

Information as practically used today is always presumed finite. Defining information in an infinite setting, turns out, is quite difficult, see e.g. Shannon Information in infinite cases.