Quasi-local approximate optimal decoders for topological quantum error correcting codes

"There has been growing interest in the classification of mixed states with topological order, particularly in how states connected via local noise channels belong to the same topological class. This concept has been applied to topological quantum error-correcting codes, where the use of the Petz recovery map has revealed that the phase transition in mixed states coincides with the decodability transition in these codes. Building on these insights, we propose a scalable serialized local decoder that provides a near-optimal approximation for decodability phase transitions. We apply our decoder to the toric code under two noise models: dephasing errors, where the decoding problem is in spatial dimensions, and a fault-tolerant setting involving both dephasing and measurement errors, where the decoding problem extends into space-time. In both cases, we present a scaling analysis, identify the phase transition, and determine the corresponding noise thresholds."