Are there any known continuous (non-lattice) quantum error correction codes?

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?

• I've not heard of codes like the ones you search for, but maybe this framework arxiv.org/abs/1309.7062 of Gottesman and Zhang might be interesting for you? Maybe this could be useful to construct codes of your type? – Martin Feb 12 '15 at 12:58
• Charges in a topological field theory like Chern-Simons theory could be used to process quantum information in a self-error-correcting way. The basic idea is that you perform unitary rotations in the ground state manifold by braiding anyons. Keeping the anyons far away from each other prevents errors. Witten's classic paper can be viewed as a prescription for a quantum algorithm to compute Jones polynomials with topological error correction built in. – Mark Mitchison Feb 12 '15 at 13:54
• The Witten paper is called "Quantum Field Theory and the Jones Polynomial" (the other link seems to have died. – Mark Mitchison Feb 12 '15 at 14:12
• Thanks Mark, I know of that paper but I never heard it interpreted as a QEC, it sounds like I should revisit it. I think that for the application I had in the back of my mind, I was hoping to find one where the information is more locally stored. And thanks Martin, I'll take a look at that one you cite. – Surgical Commander Feb 12 '15 at 17:55

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.

• +1 for the Nayak et al. review, that's a great place to learn about these topics. – Mark Mitchison Feb 12 '15 at 18:46
• Thanks for the answer and references. When you say: "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.", why do we need to have such a restrictive definition of error-correcting codes? Qubits are simple and do the job well enough, but the Hilbert spaces associated with QFT's should be able to support error-correction also, at least in principle. Could QEC be abstracted to a statement on the Hilbert space of a quantum theory? – Surgical Commander Feb 19 '15 at 10:33