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[[14,3,3]] Non-CSS Surface Code on a Rhombic Dodecahedron — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 25, 2026• 0 citations
Publisher

Abstract

FINDING: A [[14,3,3]] non-CSS surface code is realized on the rhombic dodecahedron, a space-filling polyhedron with 14 vertices, 12 faces, and 24 edges, encoding 3 logical qubits with distance 3. | MATH: Code parameters: [[n=14, k=3, d=3]]. The rhombic dodecahedron has 14 vertices (data qubits), 12 faces (stabilizers, non-CSS), 24 edges (connectivity). The stabilizer group is non-CSS, meaning X- and Z-type stabilizers are not separable; instead, they are mixed Pauli operators. The code distance d=3 implies the minimum weight of a logical operator is 3. The polyhedron's symmetry group is the octahedral group (order 48), isomorphic to S₄ × C₂. | CONNECTION: The rhombic dodecahedron is the Voronoi cell of the face-centered cubic (FCC) lattice, a root lattice of type A₃ (or D₃). Its dual is the cuboctahedron, which has edge lengths in ratio 1:√2. The FCC lattice has packing density π/(3√2) ≈ 0.74048, and its reciprocal lattice is body-centered cubic (BCC). The rhombic dodecahedron's face a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.21545252

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Rhombic Dodecahedra and Icosidodecahedra Form Radial Lattices and Quantum Surface Codes — E8 Intelligence Research

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A 67%-Rate CSS Code on the FCC Lattice: [[192,130,3]] from Weight-12 Stabilizers

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We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination K = 12); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verication conrms CSS validity (HXHT Z = 0 over GF(2)) and reveals k = 2L3 + 2 logical qubits: k = 130 at L = 4 and k = 434 at L = 6, yielding encoding rates of 67.7% and 67.0% respectively. Exhaustive search over all kernel cosets establishes the minimum distance as d = 3, independent of L. The veried code parameters are [[192, 130, 3]] at L = 4 and [[648, 434, 3]] at L = 6. This rate is 24×higher than the cubic 3D toric code (2.8% at the same distance) and, to our knowledge, the highest encoding rate reported for any three-dimensional topological stabilizer code. The high rate originates in a structural decit: the FCC lattice has 6L3 edges but only 2 ×(L3/2−1) independent stabilizer constraints, leaving ∼2L3/3 logical degrees of freedom. The minimum distance is d = 3, independent of L, established by exhaustive coset search. We provide the complete reproducible simulation code and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.

•Mar 17, 2026•0 citations
FINDING: A [[14,3,3]] non-CSS surface code is realized on the rhombic dodecahedron, a space-filling polyhedron with 14 vertices, 12 faces, and 24 edges, encoding 3 logical qubits with distance 3. | MATH: Code parameters: [[n=14, k=3, d=3]]. The rhombic dodecahedron has 14 vertices (data qubits), 12 faces (stabilizers, non-CSS), 24 edges (connectivity). The stabilizer group is non-CSS, meaning X- and Z-type stabilizers are not separable; instead, they are mixed Pauli operators. The code distance d=3 implies the minimum weight of a logical operator is 3. The polyhedron's symmetry group is the octahedral group (order 48), isomorphic to S₄ × C₂. | CONNECTION: The rhombic dodecahedron is the Voronoi cell of the face-centered cubic (FCC) lattice, a root lattice of type A₃ (or D₃). Its dual is the cuboctahedron, which has edge lengths in ratio 1:√2. The FCC lattice has packing density π/(3√2) ≈ 0.74048, and its reciprocal lattice is body-centered cubic (BCC). The rhombic dodecahedron's face a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Rhombic dodecahedra and icosidodecahedra form periodic radial-expansion lattices, with the rhombic dodecahedron also serving as a non-CSS quantum surface code. | MATH: Rhombic dodecahedron: 12 rhombic faces, each with face angle arccos(1/3) ≈ 70.53° (acute) and 109.47° (obtuse); edge length a; volume = (16√3/9)a³ ≈ 3.079a³; it is the Voronoi cell of the face-centered cubic (FCC) lattice. Icosidodecahedron: 32 faces (20 triangles + 12 pentagons), 30 vertices, 60 edges; vertex figure is a rectangle with edge ratio 1:φ (φ = (1+√5)/2 ≈ 1.618). Radial expansion of icosidodecahedra generates rhombic dodecahedral packing — the FCC lattice (coordination number 12, packing density π/(3√2) ≈ 0.7405). Quantum code: [[14,3,3]] surface code on rhombic dodecahedron (arXiv:2010.06628). | CONNECTION: The rhombic dodecahedron's face angle arccos(1/3) is the tetrahedral angle — linking to the A₃ root system and diamond cubic lattice. The icosidodecahedron's vertex figure ratio 1:φ embeds the go Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination K=12K=12K=12); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verification confirms CSS validity (HXHZT=0H_{X}H_{Z}^{T}=0HX​HZT​=0 over GF(2)) and reveals k=2L3+2k=2L^{3}+2k=2L3+2 logical qubits: k=130k=130k=130 at L=4L=4L=4 and k=434k=434k=434 at L=6L=6L=6, yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance d=3d=3d=3 is proven exactly by exhaustive elimination of all weight-≤2\le 2≤2 candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at L=4L=4L=4 and [[648, 434, 3]] at L=6L=6L=6. This rate is 24x higher than the cubic 3D toric code (2.8% at d=4d=4d=4), though at a lower distance (d=3d=3d=3 vs. d=4d=4d=4); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has 3L33L^{3}3L3 edges but only L3−2L^{3}-2L3−2 independent stabilizer constraints, leaving k=2L3+2k=2L^{3}+2k=2L3+2 logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at p=0.001p=0.001p=0.001 (and 63x at p=0.0005p=0.0005p=0.0005), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.
FINDING: The rhombic dodecahedron appears as an exact lattice structure for a quantum error-correcting surface code, while symmetry-breaking frameworks in QFT and Calabi-Yau compactifications reveal how geometric group constraints govern physical law. | MATH: Rhombic dodecahedron = zonohedron with 12 rhombic faces, 24 edges, 14 vertices; its vertices are the 12 midpoints of cube edges plus 2 cube vertices — coordinates are permutations of (±1,±1,0) and (±2,0,0) in scaled form. The [[14,3,3]] code has 14 physical qubits, 3 logical qubits, distance 3 — a non-CSS code whose stabilizer generators map to the dodecahedron's faces. Wigner-Eckart corrections under spontaneous symmetry breaking scale as O(1/N) for N→∞ systems, with Goldstone-mode contributions modifying reduced matrix elements by terms proportional to ⟨φ⟩/f (vacuum expectation value over decay constant). | CONNECTION: The rhombic dodecahedron is the Voronoi cell of the face-centered cubic (FCC) lattice — the densest sphere pack Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Surface codes (toric code) are stabilizer codes whose logical operators and error-correction thresholds are governed by the p4m wallpaper group symmetry of the square lattice, with transversal diagonal logical operators constrained by lattice automorphisms. | MATH: Toric code: stabilizers = vertex operators Av=∏e∋vXeA_v = \prod_{e \ni v} X_eAv​=∏e∋v​Xe​ and plaquette operators Bp=∏e∈∂pZeB_p = \prod_{e \in \partial p} Z_eBp​=∏e∈∂p​Ze​, with [Av,Bp]=0[A_v, B_p]=0[Av​,Bp​]=0. Logical operators: non-contractible loops ZL,XLZ_L, X_LZL​,XL​ on torus, giving k=2k=2k=2 logical qubits. Distance d=O(L)d = O(L)d=O(L) for L×LL \times LL×L lattice. Transversal diagonal logical operators: for CSS codes, diagonal gates in Clifford hierarchy Ck\mathcal{C}_kCk​ correspond to codes with certain weight conditions — Webster (arXiv:2303.15615) classifies these via code automorphism groups. p4m group: generated by translations (1,0),(0,1)(1,0),(0,1)(1,0),(0,1), 90° rotation, and two reflections; order 8 point group D4D_4D4​. | CONNECTION: p4m is the full symmetry group of the square latti Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Surface code error correction uses stabilizer formalism with parity check matrices derived from lattice geometry, enabling fault-tolerant quantum computation. MATH: - Stabilizer group S=⟨g1,g2,…,gm⟩ S = \langle g_1, g_2, \dots, g_m \rangle S=⟨g1​,g2​,…,gm​⟩ where each gi g_i gi​ is a Pauli operator (tensor product of X,Y,Z,I X, Y, Z, I X,Y,Z,I) on n n n qubits. - Parity check matrix H H H over F2 \mathbb{F}_2 F2​ (for X X X and Z Z Z errors separately) has size (n−k)×n (n-k) \times n (n−k)×n, with rows corresponding to stabilizer generators. - Surface code: qubits on edges of a square lattice, stabilizers are plaquette (Z-type) and vertex (X-type) operators. Code distance d d d scales with lattice size L L L: d=L d = L d=L for L×L L \times L L×L lattice. - Logical operators are non-trivial cycles on the dual lattice, with commutation relations given by intersection number mod 2. - Error correction threshold: pth≈0.103 p_{\text{th}} \approx 0.103 pth​≈0.103 (for standard depolarizing noise) — no simple rational ratio, but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination K = 12); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verication conrms CSS validity (HXHT Z = 0 over GF(2)) and reveals k = 2L3 + 2 logical qubits: k = 130 at L = 4 and k = 434 at L = 6, yielding encoding rates of 67.7% and 67.0% respectively. Exhaustive search over all kernel cosets establishes the minimum distance as d = 3, independent of L. The veried code parameters are [[192, 130, 3]] at L = 4 and [[648, 434, 3]] at L = 6. This rate is 24×higher than the cubic 3D toric code (2.8% at the same distance) and, to our knowledge, the highest encoding rate reported for any three-dimensional topological stabilizer code. The high rate originates in a structural decit: the FCC lattice has 6L3 edges but only 2 ×(L3/2−1) independent stabilizer constraints, leaving ∼2L3/3 logical degrees of freedom. The minimum distance is d = 3, independent of L, established by exhaustive coset search. We provide the complete reproducible simulation code and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.