Fault-Tolerant Quantum Error Correction for non-Abelian Anyons

While topological quantum computation is intrinsically fault-tolerant at zero temperature, it loses its topological protection at any finite temperature. We present a scheme to protect the information stored in a system supporting non-cyclic anyons against thermal and measurement errors. The correction procedure builds on the work of Gacs (J Comput Syst Sci 32:15-78, 1986. doi:10.1145/800061.808730) and Harrington (Analysis of quantum error-correcting codes: symplectic lattice codes and toric code, 2004) and operates as a local cellular automaton. In contrast to previously studied schemes, our scheme is valid for both abelian and non-abelian anyons and accounts for measurement errors. We analytically prove the existence of a fault-tolerant threshold for a certain class of non-Abelian anyon models, and numerically simulate the procedure for the specific example of Ising anyons. The result of our simulations are consistent with a threshold between 10(-4) and 10(-3).