Origin of the X/Z asymmetry in distributed color-code memory simulations

Determine the origin of the unexpected difference between X-basis and Z-basis logical-error curves in the distributed triangular color-code memory simulations, despite the X/Z symmetry of the color code, the superdense circuit, and the fault-improvement strategy.

Background

The distributed triangular color code is symmetric under interchange of X and Z. The simulations use a superdense extraction circuit and symmetric fault-improvement parameters, so the logical-error curves would be expected to show corresponding symmetry. Nevertheless, the reported simulations show a slight shift between the X- and Z-basis curves at code distances 7 and 9. The paper suggests that decoder behavior, including time asymmetry or detector ordering, may be responsible, but does not resolve the cause.

References

The origin of the shift is unclear, with one possibility being that it is an artifact of the decoder (either directly induced by the time-asymmetry, or originating from a random ordering of detectors that happens to disfavor the $X$ basis). We leave it as an open question.

— Resource-adaptive distributed fault tolerance with very noisy Bell pairs  (2609.03048 - Schmidt et al., 2 Sep 2026) in Section 7, subsection “Distributed Color Code”