Compare the two orientations of the Shor code

Determine which orientation of the Shor [[9,1,3]] code provides greater advantage for encoded hybrid repeater chains: the orientation with X-error protection within clusters and Z-error protection between clusters, or the orientation with Z-error protection within clusters and X-error protection between clusters, including dependence on the underlying hardware error parameters.

Background

The paper analyzes one orientation of the Shor code in which an inner three-qubit repetition code protects against X errors within each cluster and an outer repetition code protects against Z errors across clusters. The authors note that the alternative orientation reverses these protection roles and would require different fault-tolerant encoding circuits.

Because the relative frequencies of X and Z errors depend on hardware parameters, the preferable orientation may vary across platforms and operating regimes. The paper does not resolve which orientation offers the greater performance advantage.

References

We could have used the Shor code version with Z protection internally and X protection between clusters, but it is not clear which code provides an advantage. It could be true that the advantage depends on the hardware parameters, because this dictates the ratio of X and Z errors. The two versions also have different fault-tolerant encoding circuits. We leave this analysis for future work.

— Scalable entanglement distribution using encoded hybrid repeater chains  (2609.19783 - Haldar et al., 17 Sep 2026) in Appendix A, subsection “Logical Error Probabilities,” subsubsection “[[9,1,3]] Shor Code”