  2D compass codes are a family of quantum error-correcting codes that contain
the Bacon-Shor codes, the X-Shor and Z-Shor codes, and the rotated surface
codes. Previous numerical results suggest that the surface code has a constant
accuracy and coherence threshold under uniform coherent rotation. However,
having analytical proof supporting a constant threshold is still an open
problem. It is analytically proven that the toric code can exponentially
suppress logical coherence in the code distance $L$. However, the current
analytical lower bound on the threshold for the rotation angle $\theta$ is
$|\sin(\theta)| < 1/L$, which linearly vanishes in $L$ instead of being
constant. We show that this lower bound is achievable by the Z-Shor code which
does not have a threshold under stochastic noise. Compass codes provide a
promising direction to improve on the previous bounds. We analytically
determine thresholds for two new compass code families with thresholds near the
rotated surface code's numerically established coherence threshold.
Furthermore, using a Majorana mode-based simulator, we use random families of
compass codes to smoothly interpolate between the Z-Shor codes and the X-Shor
codes.
