Subset-transversal Hadamard for k>2 phantom codes

Investigate whether applying transversal Hadamard to a subset of physical qubits and/or composing such operations with qubit permutations can implement nontrivial logical Hadamard gates on CSS phantom codes with more than two logical qubits (k>2), thereby overcoming the established obstruction for global transversal Hadamard; prove or disprove the existence of such constructions.

Background

The paper shows that strictly transversal gates that do not commute with permutation-implemented logical gates are ruled out for phantom codes. In particular, a global transversal Hadamard cannot implement logical Hadamard for k>2 phantom codes. However, it is unclear whether weakening to subset-transversal Hadamard (acting on a subset of qubits) and/or composing with permutations could circumvent this barrier.

Resolving this would clarify whether ZX-duality (balanced X/Z sectors and logical Hadamard structure) is achievable for phantom codes beyond k=2 and would illuminate deeper constraints on their transversal logical gate sets.

References

We suspect that phantom codes with ZX-duality for k > 2 may not exist. One can show that transversal H{\otimes n} implementing logical Hk up to any logical CNOT for k > 2 is not possible using Theorem ... and Lemma ..., but it is not clear whether transversal H on a subset of qubits and/or composed with permutations that are not necessarily a valid logical operation can overcome this barrier.

— Entangling logical qubits without physical operations  (2601.20927 - Koh et al., 28 Jan 2026) in Appendix (Phantom quantum Reed–Muller codes), subsection 'Construction'