Decoding Topological Subsystem Color Codes Over the Erasure Channel Using Gauge Fixing

Topological subsystem color codes (TSCCs) are an important class of topological subsystem codes that allow for syndrome measurement with only 2-body measurements. It is expected that such low complexity measurements can help in fault tolerance. While TSCCs have been studied over depolarizing noise model, their performance over the erasure channel has not been studied as much. Recently, we proposed erasure decoders for TSCCs and reported a threshold of 9.7%. In this paper, we continue our study of TSCCS over the erasure channel. We propose two erasure decoders for topological subsystem color codes. These decoders employ a mapping of the TSCCs to topological color codes (TCCs). In addition, these decoders use the technique of gauge fixing, where some of the gauge operators of the subsystem code are promoted to stabilizers. We perform gauge fixing using 4-body and 8-body gauge operators. With partial gauge fixing, we obtained a threshold of 17.7% on a TSCC derived from the square octagon lattice. Using an order maximal gauge fixing decoder we were able to improve the threshold to 44%. The performance of the order maximal gauge fixing decoder can be further improved to close to 50% in conjunction with an optimal erasure decoder for topolological color codes. We also study the correctability of erasures on the subsystem codes.