I come from a hep-th background, but I have noticed that quantum information is becoming increasingly common in discussions of AdS/CFT and black hole information, and so I've begun thinking about it lately. I'm heard much about the toric code and other quantum error correction codes, and I'm wondering if a continuous, non-lattice version of these codes exists. It seems certain that such a thing should exist in principle, but are there are known models, for example one based on a relativistic field theory?
For one thing, toric codes (and other error-correcting codes) are really about ways to store quantum information(logical qubits) in several physical qubits, so there is not much point in asking for a continuous limit. On the other hand, if you view it as a topological quantum phase of matter, then surely there are continuous versions. For example, the topological order of the toric code can be described by a Abelian Higgs theory(i.e. $U(1)$ gauge field coupled to charge-$2e$ matter). There are of course, generally Chern-Simons theory, as pointed out in @Mark Mitchison's comment. The idea of topological quantum computation(TQC) is to store quantum information in the degenerate non-local Hilbert space of multiple non-Abelian anyons, and use braiding to perform gate operations. But notice the difference between TQC from the toric code, since as a topological phase toric code only supports Abelian anyons. The logical qubit is related to the degenerate ground states when the toric code is placed on a torus, if I understand correctly.
Witten's paper is of course a classic on Chern-Simons theory. For the application to TQC this review(http://arxiv.org/abs/0707.1889) is a good place to start.