Fault tolerance of the superdense color-code extraction circuit

Prove fault tolerance of the superdense stabilizer-extraction circuit for the triangular color code, including whether it preserves the full code distance under circuit-level noise.

Background

The paper discusses the superdense circuit as an alternative to Bell-flagging for measuring both X- and Z-type color-code stabilizers. Although its functional validity follows from ZX-calculus, the Hopf rewrite used in its construction is not fault-equivalent. Existing numerical and exact-search evidence supports full-distance behavior for several small code distances, but does not establish the result generally. Consequently, the circuit’s fault tolerance remains unresolved, which is particularly important because the superdense construction can reduce the number of inter-QPU distribution edges by a factor of two.

References

However, the Hopf rewrite is not a fault equivalent rewrite, and the fault tolerance of the superdense circuit remains an open question.

— Resource-adaptive distributed fault tolerance with very noisy Bell pairs  (2609.03048 - Schmidt et al., 2 Sep 2026) in Section 6, subsection “Syndrome Extraction and Benign Hook Errors in the Color Code”