Fig. 8: MWPM-based patch decoder.

a Example of an error chain which creates four e-anyons. b The decoding algorithm performs correction using only local information by splitting the large system into smaller overlapping regions, within each of which the MWPM algorithm is used to find the lowest-weight pairing of anyons. These local regions have open boundaries, hence MWPM can also pair anyons to the boundaries if this is of lower weight. In practice, a slight boundary bias is added to break ties in favor of boundary pairing. c The final step requires locally combining the pairing outputs to determine the final pairing. In particular, we count the number of times each site p is paired to sites q > p. In the diagram, two equal-weight pairings contribute 0.5 each, though we randomly break the tie in practice. Then, the algorithm pairs p with the q that appears most often. In this example diagram, we connect two pairs that have weight = 5.5, and do not form the weight = 2 pairing. We see in the simple four-anyon case depicted above, the procedure correctly recovers the pairing with windows of size l = 3. In general, this patch-based decoder can correct errors up to distance d = ⌊l/2⌋; moreover, the distance by which it spreads information and the thickness of any associated LED operators are both proportional to l.