Fundamental physics has a strong appeal to the imagination. There ought to be some underlying structure to our theories of physics, something beautiful and intuitive that explains the parameters of the standard model, unifies gravity and quantum mechanics, explains puzzles like the black hole information paradox, and probably has a few bonus surprises in store for us. This is probably one of the main reasons many people with strong expertise in diverse disciplines take an active interest, read blogs and popular books on the subject, and sometime try to lend a hand and help.
It is natural for a person with an intimate knowledge of some theoretical structure to try to apply it to the main few questions of fundamental physics. Could this fundamental theory involve spin chains, Turing machines, cellular automata, or your favorite (sort of) Lie super algebra? This looks much more likely if you devoted your career to studying the intricacies of these structures. There is of course nothing wrong with that, nobody has any idea what the fundamental theory of our universe looks like, the issues are difficult and we can use all the help we can get. To facilitate such help I’ll offer some unsolicited advice: Nature is relativistic, and this fact is crucially important!
In fact, Lorentz invariance is so important, and relativistic systems are so entirely different from non-relativistic ones, that there is whole discipline (theoretical high energy physics) which is devoted almost exclusively to the study of Lorentz invariance and its consequences. Along the way we stumbled upon a few things that may possibly be interesting and relevant in our search for that underlying structure.
The message has two parts really, both need some unpacking so I’ll only summarize them here. First, fundamental violations of Lorentz invariance, meaning the idea that Lorentz symmetry is not an ingredient of that holy grail, the fundamental theory of our universe, are strongly excluded by experimental constraints. It is not known how to break Lorentz symmetry in a way that induces only small effects on observable physics (unlike for example symmetries like baryon number that can be violated by a small amount; technically: there are many new relevant and marginal operators once you allow for Lorentz violations at high energy, their coefficients are constrained to a ridiculous degree by experiment). Therefore, all the evidence we have indicates that the fundamental theory is Lorentz invariant.
(That is not to say that Lorentz invariance cannot be violated spontaneously. Lorentz invariance is the symmetry of empty space, and in our universe space is not really empty. However, in that case the equations governing the dynamics are still symmetric, the physics at high energies (or short distances) is insensitive to the breaking, so most of the consequences of the symmetry are still there).
Secondly, on the more theoretical side, Lorentz invariant systems are almost impossible to quantize. It took a series of miracles over a few decades to find a consistent quantum mechanical Lorentz invariant theory, namely quantum field theory. In other words, the demand of Lorentz invariance, made impossible to ignore by experimental constraints, is highly restrictive theoretically. For example, there is a whole slew of no-go theorems (which go under such names as Weinberg-Witten or Coleman-Mandula), but also common lore considerations and just common sense arguments that pretty much exclude many models of fundamental physics from the get go, without having to get into any details.
For the innovative outsider, this message brings both good news and bad news. The good news are that if you ignore the restrictions coming from Lorentz invariance (or better still, if you manage to find a convincing loophole), the classic open problems of fundamental physics become wide open. Lots of ideas that are not even entertained by the community of experts (or have long been dismissed by them, as the case may be) can suddenly become relevant. It is likely you can come up with many wonderfully creative insights, write papers and popular articles, and make some real interesting contributions. The bad news are left as an exercise to the reader.
Oh yeah, almost forgot, that business with the computer in the title… As is probably clear by now, this post is a bait and switch kind of post; it is not really specific to computers, quantum or otherwise. In fact, as far as I know, as long as you don’t define precisely what you mean by a computer (something that some advocates of the idea carefully avoid), the universe could very well be a computer (same goes for my socks) . But serious people would probably have a computational model in mind, something like a Turing machine or the standard circuit model of quantum computing. In any such a model, the first thing I would try to understand is how Lorentz symmetry is implemented.
This should not be simple, as there is some tension in my mind between existing models of computation and Lorentz invariance. Namely, in the Lorentz invariant theory the space of states is always continuous. In technical Language the Lorentz group is non-compact and all its representations are continuous. In plain English: you can boost any state to have an arbitrary momentum, which can be any real number. Any fundamental discreteness, such as is common in all computational models I know of, would have to be of a very special sort to hide its Lorentz violating effects from existing tests of special relativity (no need for any new tests). Perhaps this is possible, but how this can be done is a mystery to me, and so this is to my mind the first serious test of any discrete model purporting to be a theory of nature.
While we are at it, maybe starting small would be more appropriate – let’s forget about everything and concentrate on something, especially something we already know well. How about quantum electrodynamics, the theory of photons and electrons, the familiar playing ground where we know how to calculate pretty much anything, can that theory be efficiently simulated by your favorite universal model of quantum computation? I for one would be interested if it did.