[[14,3,3]] rhombic dodecahedron surface code

Direction & hypothesis

Twist-defect surface codes are the non-CSS corner of the surface-code family: the qubits sit on a polytope's vertices and the twists force generators that mix X and Z. The Error Correction Zoo (https://errorcorrectionzoo.org) publishes the full stabilizer tableau for one of them together with its parameters, which is enough to test whether the point records anywhere. The answer here is conditional in an instructive way — the same parameters advance only in the non-CSS form — so it also says something about where the board is thin.

Source: the zoo entry rhombic_dodecahedron_surface (https://errorcorrectionzoo.org/c/rhombic_dodecahedron_surface), the [[14,3,3]] twist-defect code on the vertices of a rhombic dodecahedron, whose tableau is Landahl's (arXiv:2010.06628, Eq. 1). The zoo notes a local-Clifford-equivalent clean realization that is CSS on its four-valent vertices.

What was searched

A crawl of the zoo index filtered to entries whose hierarchy path passes through a QLDPC concept returned 681 quantum entries; 190 are primary qLDPC nodes, of which 35 state a three-parameter [[n,k,d]] claim. Each claim was bracketed against a snapshot of the live board before any matrix was built — exact match on (n,k,d), dominated in every cell, records in some cells, or records in all of them. This claim was one of two in the conditional bucket: it records if its check weight and board type put it in an open cell, and it does not otherwise. No construction search was run; the generators came from the zoo.

Evidence trail

The verifier's Pauli-weight witness search (20,000 random information-set trials, then the 2,000,000-trial accelerator pass on the symplectic doubling) returned a witness of Pauli weight 3, so the claim is d <= 3, a witness-backed upper bound matching the zoo's stated distance. Structural verification passed: all eleven generators mutually commuting (rank 11 on 14 qubits, so k = 3), max check weight 4, witness valid.

The trusted gate verify/validate_candidate.py on the built document returned passed: true with the label "advances the weight-4 x unrestricted stabilizer board", no exact duplicate and no WL-equivalent entry on either board. The conditional bracketing resolved as follows: as the non-CSS tableau given here it lands on the stabilizer board, which is open; had it been submitted as a CSS code, the weight-4 cell already contains [[12,3,3]], which would dominate it. Literature novelty is reported as unverified.

Dead ends

Of the 35 stated claims, 20 were already on the board as exact parameters and 9 were dominated in every cell they could land in. The other conditional claim, the [[30,8,3]] Bring code, has weight-5 generators, which puts it in the weight-6 class where [[25,9,3]], [[30,10,3]] and [[30,8,4]] already dominate it; it was not built. Submitting the local-Clifford-equivalent CSS form of this code would also have been dead on arrival for the same reason as above.

Tools

Model Mimo-V2.6-Flash; the repository CLI for the build, the witness search and the submission document, and the repository's verify/ stack for the gate. Approximate compute: under a minute of CPU.

Reproduction

The zoo tableau, eleven generators on fourteen qubits:

X X X I I X I I I I I I I I I I X X I I X X I I I I I I I I I I I I I I X X I I X X X I I I I I I I I X I I X X I Y Y Y Y I I I I I I I I I I I Y I I Y Y I I I Y I I I I I I I Y I I I I Y Y I I Y I I I I I I I I I I Y Y Y Y Z Z I I Z I I I I Z I I I I I I I Z Z I I Z Z I I I I I I I I I I I Z Z I I Z Z I I

Each row becomes one row of S = (A | B), with Y setting a 1 in both halves. The CLI takes an .npz with keys a and b, or a single key s holding A | B. n = 14, rank S = 11 so k = 3, max check weight 4, distance witness 3. The invocation that produced codes/14-3-3.json was ./qldpc submit with --authors @MathysRennela --model "Mimo-V2.6-Flash" --family topological.

Cite the source: "([[14,3,3]] Rhombic dodecahedron surface code)", The Error Correction Zoo (V. V. Albert & P. Faist, eds.), https://errorcorrectionzoo.org/c/rhombic_dodecahedron_surface, arXiv:2606.11484; tableau from A. J. Landahl, "The surface code on the rhombic dodecahedron", arXiv:2010.06628.