# Code distance and other questions about Quantum double model as an error correcting code

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?

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.

• 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? Dec 5, 2021 at 10:06
• 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. Dec 5, 2021 at 14:47
• 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. Dec 5, 2021 at 14:49
• sounds interesting what is the paper by Bombin and Haah? Dec 5, 2021 at 16:51
• Haah's paper: arxiv.org/abs/1812.11193 Dec 5, 2021 at 20:42