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We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size l, our algorithm runs in time logl compared to l6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.
Duclos-Cianci et al. (Fri,) studied this question.