So this is more a quantum information/computation theory sort of question, but let me try and phrase it the best I can:
An algorithmic computation between two states (ie bitstrings) - any computation - can be performed with a small set of gates, which in QIT, means rotations in Hilbert space. If our set of data is the state of some quantum field, where qubits may be as simple as true-false statements about particle eigenstate existence, or as complex as higher n-ary number states represented by various degeneracies; can we still represent the total algorithmic complexity involved with some small set of "gates" (ie ladder operators) or is quantum field theory not capable of such a feat?
@Nikolay: Yes, this is what I described in the first sentence of my second paragraph. Sorry if I made it too vague, I was trying to generalize the idea of that phase operation, which is really the core of quantum computation.
To be clear: I'm looking for some way of describing particle interactions (electric, magnetic flavor, color, etc...) in terms of purely these phase operations. Essentially, a molecule-molecule interaction would represent some set of computations performed, the results of which are contained in the daughter molecules' quantum state. A good model here describes gate operations in a manner similar to how they actually occur physically, where a finite set of our usual gates might have a hard time emulating the full complexity of that interaction with respect to an external weakly interacting environment.
That is to say...I know we can build a house with nothing but 2x4's, hammers and nails, but if we have access to every tool in existence, even ones we normally wouldn't ever think of using for building a house (like the knowledge of tree-shaping, idk) then how much does that let us simplify algorithms?
Edit: actually, I'm looking at one of your RG publications (Encoding and decoding of additional logic in the phase space of all operators) and a lot of the mathematics you have laid out fit pretty well with what I'm trying to do. Gotta get to bed right now, but I'll read some more ASAP tomorrow.
Indeed it is-the gates are unitary transformations and what must be taken care of is that there aren't any cocycles, that express anomalies.
@Stam: I will look into cocycles, I'm not quite sure how they fit into this though as I've never even heard the word. (if you could give me a brief explanation that'd be awesome)
And Nik: Sorry I haven't had a chance to reply, I broke my collarbone Friday night and have been loopy on pain meds this whole time. Plus I can't use my right (primary) hand so it's making the thinking/working process a little different than usual!
Cocycles is the mathematical term and a nice paper is this one: http://www.sciencedirect.com/science/article/pii/0370269385911463
The way they fit into the discussion is the following: the ``quantum gates'' ,that take qubits as inputs and give other states of qubits as outputs, are unitary maps. They must realize transformations in such a way that, when using two different sets of transformations on the same inputs, on should get the same output, if the two sets are equivalent. The cocycle condition expresses this equivalence. For an example cf. http://arxiv.org/abs/math-ph/9805012 and http://arxiv.org/abs/1411.7293 . It's the prefactor that makes a projective representation a faithful representation.
Hmm ok I think I have an idea of this now; the papers involved are a bit beyond me mathematically right now (many of the group theory concepts are ones I haven't read of yet) but from what I can tell this is exactly what I'm looking for.
Thanks!
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