Exactly four twelve-qubit constant-excitation CSS codes
Abstract
A quantum code is constant excitation (CE) if every computational basis ket occurring in its code space has the same Hamming weight; such a code acquires only a global phase under collective coherent Z-rotations. Lai, Liou and Ouyang generalise the CSS construction to produce CE codes by adding a global shift and exhibit a [[12,1,3]] example, and twelve is the least length at which a CE CSS code with one logical qubit exists. We classify that length. Up to permutation of the qubits — and up to the redundancy in the global shift, which is a freedom in the presentation rather than a symmetry — there are exactly four [[12,1,d ≥ 3]] CE CSS codes. We exhibit the four: their X-type codes C₁ have dimensions 5, 4, 4, 3 and weight enumerators 0¹4¹⁵8¹⁵12¹, 0¹4⁷8⁷12¹, 0¹4³6⁸8³12¹ and 0¹4³8³12¹, and all four have d_X = 4 and d_Z = 3, so the distance is decided by d_Z in every class. The weight enumerator of C₁ is an invariant of the equivalence, which is what makes the four provably four; the code of Lai, Liou and Ouyang lies in the first class. Two structural by-products: no [[11,1,d ≥ 3]] CE CSS code exists, and this now has a proof that runs no search at all, so the most expensive computation of the earlier nonexistence work is no longer needed; and every [[13,1,d ≥ 3]] CE CSS code is a padding of a twelve-qubit one. The classification rests on one exhaustive computation: at the single shift the structure theory leaves, 46 350 pairs (C₁,φ) survive the distance conditions, and each carries a permutation certificate identifying it with one of the four. Every theorem below is machine-checked in Lean 4; the exceptions — the passage from the quantum-mechanical definitions to the finite ones, and two clauses in the exhibition of the four codes that the classification itself does not use — are marked where they occur.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
CC1known2026-08-30
The excitation of a CE CSS code is forced: if C1 has full support on n coordinates then n = 2*wt(y), so a full-support CE CSS code has even length
CC2routine2026-08-30
Shortening: a coordinate outside supp(C1) may be deleted, so at odd length every CE CSS code is a padding of a shorter one; hence no [[11,1,d>=3]] CE CSS code, from no_codeₜen, with no search
CC3routine2026-08-30
Permutation equivalence of CE CSS codes: CEquiv (qubit permutation + the shift's freedom modulo C1) is an equivalence relation, carried by transposition lists; Code.rebase; and the weight enumerator of C1 is an invariant
CC4routine2026-08-30
The n = 12 classification is a single-shift problem: every [[12,1,d>=3]] CE CSS code has full support, hence wt y = 6, hence is equivalent to one with y = 63
CC5routine2026-08-30
The four representatives Ka, Kb, Kc, Kd: genuine [[12,1,3]] CE CSS codes with d_X = 4 and d_Z = 3 exactly, C1 weight enumerators 0¹ 4¹5 8¹5 12¹, 0¹ 4⁷ 8⁷ 12¹, 0¹ 4³ 6⁸ 8³ 12¹, 0¹ 4³ 8³ 12¹, pairwise inequivalent
CC6routine2026-08-30
Soundness of the classifying search: a permutation certificate on the labelled subspace 2v + phi(v) is an equivalence (labelledₑq, cequivₒf_cert), and the enumeration reaches every code's node (dfsCₛound)
CC7candidate2026-08-30
Headline: there are EXACTLY FOUR [[12,1,d>=3]] constant-excitation CSS codes up to qubit permutation and the shift's freedom – every such code, any shift and any construction, is equivalent to one of Ka, Kb, Kc, Kd, and those four are pairwise inequivalent
CC8known data2026-08-30
Controls: the certificate table rejects a non-accepting state and accepts the source's own code; a certificate must name the right class; CEquiv is neither the identity relation (Ka.swap 0 1 has a different C1) nor everything (the four are inequivalent); the shift clause is exercised by Code.rebase
CC9routine2026-08-30
The source's own [[12,1,3]] code (LeanProblemSpec.CeCss.codeTwelve, Appendix A.6) is in class Ka, the dim C1 = 5 class of orbit size 7200; and every [[13,1,d>=3]] CE CSS code is a padded twelve-qubit one, so no n = 13 search is needed
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- There are exactly four [[12,1,3]] constant-excitation CSS codes. Up to permutation of the twelve qubits — and the freedom in the global shift, which is not a choice but a redundancy in the presentation — every [[12,1,d ≥ 3]] CE CSS code is one of four, and those four are pairwise inequivalent.
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- 2026-09-07 03:53 UTC
- Ledger commit
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