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What, exactly, makes the toric code a quantum error-correcting code as opposed to any other string-net model? What makes it special? The way I understand it, it's a normal string-net model on a torus, where the stabilizer space for such a model is 4D and thus it can support two qubits. Is this space 4D because of the periodic boundary conditions on the torus?

Also, I am a little confused what the toric code tells us. Is it a static theory of anyons, or can we simulate braiding statistics?

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    $\begingroup$ The four-fold degeneracy does come from periodic boundary conditions. For your second question, the toric code is exactly solvable. You can of course calculate the braiding statistics of anyons, even analytically, since you know the string operators that create and move anyons. This is how we knew there are anyons in this model in the first place. $\endgroup$
    – Meng Cheng
    May 4, 2015 at 23:05
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    $\begingroup$ Any string-net model can be used as an error correcting code. The Toric Code is simply the easiest to realize (qubits, four-body stabilizers, simplicity of encoding/correction). $\endgroup$
    – Norbert Schuch
    May 7, 2015 at 4:12

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