The quantum erasure channel is used to model the phenomenon of qubit loss which occurs in many quantum technologies. Unlike the depolarizing channel, it is much more tractable. A qubit loss can be detected, and the location of the qubit can be identified. Also, the performance of a quantum error-correcting code under the erasure channel often gives insights into its performance for general channels. This channel has been studied extensively for various classes of codes like quantum BCH codes, surface codes, topological color codes to name a few. But surprisingly, an important class of codes called the topological subsystem color codes (TSCCs) has not been studied yet. These codes have a simpler syndrome measurement process in comparison to the stabilizer codes. Particularly they involve only 2-body measurements, which help in fault tolerance.

Motivated by this, we study the performance of TSCCs over the erasure channel. It is the first such study for this problem. We propose multiple decoding algorithms with or without gauge fixing. In gauge fixing, we further constrain the codespace by fixing some additional degrees of freedom available in subsystem code. We propose three decoding algorithms without gauge fixing, and the best of them gives us a threshold of 9.7%. To improve the performance, we use partial gauge fixing, which enhances the performance to 17.7%. By extending it to full gauge fixing, we achieve a remarkable improvement in the performance to obtain a threshold of 44%. We focus on TSCC derived from the square octagon lattice, but our ideas can be adapted for general subsystem codes