Logical Error Rate of 2D Compass Codes Under Small Gauge Changes for Stochastic Noise

2D Compass Codes are topological stabilizer quantum error correction (QEC) codes that can correct both bit-flip (X) and phase-flip (Z) errors by using local stabilizer generators, leveraging the spatial structure of qubits on a lattice. Within the family of 2D Compass Codes, codes can be transformed into one another by adjusting the spatial distribution of X and Z checks. We refer to each of these transformations as gauge changes, indicating the subsystem structure of 2D Compass Codes. In this work, we present a method to quantify the impact of small gauge changes on the logical error rate by analyzing the characteristics of weight enumerator combinations. This approach enables the establishment of upper and lower bounds on the logical error rate of 2D Compass Codes under small gauge changes, providing an effective method to characterize the performance of 2D Compass Codes that differ by only a few gauge changes.