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Minimum-weight perfect matching (MWPM) has been been the primary classical algorithm for error correction in the surface code, since it is of low runtime complexity and achieves relatively low logical error rates [Phys.
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As the error correction algorithm of Ref. Duclos-Cianci and Poulin 2010a itself allows for trade-offs between runtime and error rates, and MWPM based algorithms have attained most interest in the literature, we do not directly compare our algorithm to the former
Cited in the paper.
If there are no virtual anyons on a certain boundary, we connect the additional virtual anyon to all real anyons of the same type, with edges weighted by their distance to the corresponding boundary. If there are no real anyons, we connect the two new virtual anyons placed on opposite boundary. This way, the minimum weight error configuration of each equivalence class can be found for any possible anyon configuration
Cited in the paper.
Neglecting, again, O ( log L ) O(\log L) factors due to the random number generator which is necessary to propose one of the O ( L 2 ) O(L^{2}) stabilizer operators for the Metropolis procedure
Cited in the paper.
2013
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