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Kitaev's quantum double model is an error correcting code, see:

https://arxiv.org/abs/1908.02829

I am in a class on quantum error correction and the professor commented that a quantum double model for any non abelian finite group is an example of a quantum error correcting code which is not a stabilizer code (and thus certainly not a CSS code). Does that answer this question:

Topological quantum error correcting codes which are not CSS codes

Anyway, I am curious about the code parameters of the quantum double model for finite group $ G $ and a lattice with $ E $ edges on a surface $ \Sigma $ with genus $ g $.

The dimension of the physical Hilbert space is $ N=|G|^E $ where $ |G| $ is the number of elements in the finite group $ G $. And $ K $, the dimensions of the code space, is the number of orbits of $ Hom(\pi_1(\Sigma),G) $ under the action of conjugation by $ G $ ( theorem 2.4 of the link above). When $ G $ is abelian this simplifies significantly to $ K=|G|^{2g} $.

What is the distance $ d $ of the code? Is it the same as for the Toric code? Namely, $ d $ is the number of lattice edges of the shortest noncontractible cycle on the lattice or dual lattice?

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The logical operators for Kitaev's quantum doubles are the ribbon operators, They are roughly speaking still defined on non-contractible paths on the lattice. So the code distance scales in the same way with the lattice parameter as the toric code.

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  • $\begingroup$ Wow I saw you talk at a conference once so cool to have you answer my question here!! What is your intuition about the quantum double model for a non abelian finite group not being equivalent to a CSS or stabilizer code? Also (maybe this is a very obvious) but is it true that the logical operators of the code (the non contractible ribbons) are not even unitary? $\endgroup$
    – Ian Gershon Teixeira
    Dec 5, 2021 at 10:06
  • $\begingroup$ There is a very general theorem (by Bombin and Haah) that 2D CSS code (or even more general Pauli stabilizer code) can always be transformed into stack of toric codes + some trivial stuff. Since the topological order of Kitaev's quantum double is that of a discrete non-Abelian gauge theory, it can not be a Pauli stabilizer code. $\endgroup$
    – Meng Cheng
    Dec 5, 2021 at 14:47
  • $\begingroup$ About the ribbon operators, in their original form they are non-unitary. This is the price we pay in order to write them down explicitly. $\endgroup$
    – Meng Cheng
    Dec 5, 2021 at 14:49
  • $\begingroup$ sounds interesting what is the paper by Bombin and Haah? $\endgroup$
    – Ian Gershon Teixeira
    Dec 5, 2021 at 16:51
  • $\begingroup$ Haah's paper: arxiv.org/abs/1812.11193 $\endgroup$
    – Meng Cheng
    Dec 5, 2021 at 20:42

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