Extend magic state distillation to other permutation-invariant code families

Determine whether families of permutation-invariant quantum codes other than gnu codes can be used to implement the same projection-and-decoding magic state distillation procedure for single-qubit states and ascertain whether such protocols achieve behavior and performance comparable to or better than the gnu-code-based protocol (e.g., in error suppression and overall results).

Background

The paper introduces a magic state distillation protocol based on permutation-invariant gnu codes, demonstrating high error thresholds and flexibility with as few as two qubits. The method projects multiple noisy copies of a single-qubit state onto the gnu codespace, decodes to a single qubit, and post-selects on success.

Because permutation-invariant codes form a broad family beyond gnu codes, a natural next step is to investigate whether other permutation-invariant code families can realize similar or improved distillation behavior under the same protocol. This would clarify how general the observed advantages are and whether further gains are possible by exploring different code constructions.

References

Several open questions remain for future investigation. While we have restricted our analysis here to gnu codes, we expect that similar behaviour may arise for other families of permutation-invariant codes, we would like to investigate whether these other families of codes can provide similar or possibly enhanced results.

— Magic state distillation with permutation-invariant codes and a two-qubit example  (2603.04310 - Leitch et al., 4 Mar 2026) in Section: Discussions (Section 5)