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

Abstract

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 verification confirms CSS validity ($H_{X}H_{Z}^{T}=0$ over GF(2)) and reveals $k=2L^{3}+2$ logical qubits: $k=130$ at $L=4$ and $k=434$ at $L=6$, yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance $d=3$ is proven exactly by exhaustive elimination of all weight-$\le 2$ candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at $L=4$ and [[648, 434, 3]] at $L=6$. This rate is 24x higher than the cubic 3D toric code (2.8% at $d=4$), though at a lower distance ($d=3$ vs. $d=4$); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has $3L^{3}$ edges but only $L^{3}-2$ independent stabilizer constraints, leaving $k=2L^{3}+2$ logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at $p=0.001$ (and 63x at $p=0.0005$), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.

Figures (2)

• Figure 1: Structural comparison. (a) The FCC lattice at $L = 4$: 32 nodes, 192 edges (= physical qubits), $K = 12$ per node. (b) The cubic lattice at $L = 4$: 64 nodes, 108 edges, $K = 6$. (c) The rate difference explained: the FCC lattice has 192 edges but only 62 stabilizer constraints, leaving 130 logical qubits (green). The cubic lattice has 108 edges but 105 constraints, leaving only 3. (d) Octahedral void with its 6 surrounding nodes and 12 edges: the X-stabilizer. (e) Tetrahedral void with $c = 3$ skew-edge pairs (coloured): the logical qubit. (f) ABC stacking of hexagonal layers for hierarchical decoding.
• Figure 2: The FCC $K = 12$ CSS code. (a) Encoding rate comparison: the FCC code achieves $67.7\%$ (at $d = 3$), compared to $2.8\%$ for the cubic toric code (at $d = 4$). (b) Computationally verified parameters. (c) Origin of the high rate: $3L^3$ edges minus $L^3 - 2$ stabilizer constraints leaves $k = 2L^3 + 2$ logical qubits (green region). (d) Asymptotic rate approaches $2/3$. (e) Rate-distance tradeoff: the FCC code occupies a new point in the landscape.

Theorems & Definitions (1)

• Definition 1: FCC Stabilizer Code