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arXiv:2603.20294

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

Raghu Kulkarni

quant-phSubmitted Mar 19, 2026• Announced Mar 20, 2026• 0 citations
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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.

Comments
14 pages, 2 figures, 3 tables, 2 appendices with reproducible Python code

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

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The [[3L3, 2L3+2, 3]] CSS code on the Face-Centered Cubic (FCC) lattice achieves 67.7% encoding rate with weight-12 stabilizers, but its distance is capped at d = 3 by weight-3 logical operators sitting at tetrahedral interstitial voids. These logicals carry a geometric 􏰂ngerprint􏰀one error per triad sheet, all three converging at a known void􏰀that makes them detectable by a 􏰃ag ancilla measuring the three-body parity Zea Zeb Zec . The weight-3 logical 􏰃ips this parity from +1 to −1; the stabilizer syndrome stays silent, but the 􏰃ag sees it. Running R independent 􏰃ag rounds per syndrome cycle pushes the logical failure rate to p3+R, so the e􏰁ective distance is simply deff = 3 + R􏰀adjustable at runtime, with no changes to the code or its logical count. At R = 4 the FCC+􏰃ags code suppresses errors as well as a distance-7 surface code, while encoding 130 logical qubits at 29.0% rate versus 1 logical qubit at 1.0%. An adaptive scheduling protocol cuts the 􏰃ag overhead to under 0.1% of measurement cycles at realistic noise levels.

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Abstract We present a CSS quantum error correcting code on the Face-Centered Cubic (FCC) lattice that combines surface-code-compatible K=4 connectivity with native inter-sheet logical gates between co-located CSS code blocks — a combination not previously achieved among well-studied codes. The construction has two parts. First, restricting the [[3L³, 2L³+2, 3]] FCC lattice code to a single triad sheet yields the sheet code with parameters [[L³, 2L, L]] at even L (or [[L³, L, L]] as a planar variant), uniform weight-4 stabilizers, and K=4 per-qubit connectivity. Three triad sheets share the FCC lattice geometry and can be deployed in one of three ways — monolithic 2D for small L, a three-layer stacked architecture for scalable deployment, or native 3D hardware — in each case encoding 6L logical qubits at distance L. Second, we introduce a fault-tolerant lattice surgery protocol that uses local FCC triangle measurements to implement joint Pauli measurements between logical qubits in different sheets. Each surgery operation adds O(L) two-qubit gates to the O(L³) per-round syndrome extraction cost. Under circuit-level depolarizing noise with MWPM decoding, the static memory threshold is p_th ≈ 0.63% and the surgery threshold is p_th ≈ 0.5%, both above the operating range of current quantum processors. Compared to surface code lattice surgery (also K=4, lattice surgery confined within a single patch), the sheet code provides 2L logicals per sheet at the same distance and connectivity; compared to bivariate bicycle codes (which target inter-block gates between physically separated modules at K=6 with long-range couplers), the sheet code achieves inter-sheet gates at K=4 with only short-range couplings between co-located sheets.

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Quantum XYZ Stabilizer Codes

Alessio Baldelli, Davide Orsucci, Francisco Lázaro +1 more

Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.

cs.ITCross-listedarXiv:2607.14988•Jul 17, 2026•0 citations
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.
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.
The [[3L3,2L3+2,3]][[3L^3, 2L^3{+}2, 3]][[3L3,2L3+2,3]] CSS code on the Face Centered Cubic (FCC) lattice reaches a 67.7% encoding rate with weight-12 stabilizers. Its distance is 3. We classify every weight-3 logical operator of the code. The X-type ones are the 4L34L^34L3 triangular faces of the lattice. The Z-type ones are the 4L34L^34L3 claws, each formed by the three edges of a tetrahedron meeting at one of its vertices. The two families are exact duals under the exchange of vertices and octahedral voids, and each spans 2L3−12L^3 - 12L3−1 independent logical classes. We prove this count analytically from the homology of the 3-torus, not only numerically: the cycle space has dimension 52L3+1\tfrac{5}{2}L^3 + 125​L3+1 , the boundary space has dimension 52L3−2\tfrac{5}{2}L^3 - 225​L3−2 , and subtracting the stabilizer rank leaves 2L3−12L^3 - 12L3−1 for every even L. So k=(2L3−1)+3k = (2L^3 - 1) + 3k=(2L3−1)+3 , and the excess rate is carried entirely by weight-3 operators. We then show that this sector cannot be protected. The three-body parity ZeaZebZecZ_{e_a}Z_{e_b}Z_{e_c}Zea​​Zeb​​Zec​​ at a face anticommutes with four octahedral stabilizers, so it is not measurable, and no measurement of any kind can distinguish ∣ψ⟩{|{\psi }\rangle }∣ψ⟩ from L^T∣ψ⟩\hat{L}_T{|{\psi }\rangle }L^T​∣ψ⟩ for arbitrary encoded ∣ψ⟩{|{\psi }\rangle }∣ψ⟩ . Removing the sector requires adjoining all 2L3−12L^3 - 12L3−1 independent directions, which returns k=3k = 3k=3 . Verified exactly over GF(2) at L=4,6,8,10L = 4, 6, 8, 10L=4,6,8,10 .
Abstract The [[3𝐿 3 , 2𝐿 3 +2, 3]] CSS code on the Face-Centered Cubic (FCC) lattice achieves 67.7% encoding rate with weight-12 stabilizers, but its distance is capped at 𝑑 = 3 by weight-3 logical operators sitting at tetrahedral interstitial voids. These logicals carry a geometric fingerprint—one error per triad sheet, all three converging at a known void—that makes them detectable by a flag ancilla measuring the three-body parity 𝑍 𝑒𝑎 𝑍 𝑒𝑏 𝑍 𝑒𝑐 . The weight-3 logical flips this parity from +1 to −1; the stabilizer syndrome stays silent, but the flag sees it. Running 𝑅 independent flag rounds per syndrome cycle pushes the logical failure rate to 𝑝 3+𝑅 , so the effective distance is simply 𝑑eff = 3+𝑅—adjustable at runtime, with no changes to the code or its logical count. At 𝑅 = 4 the FCC+flags code achieves the same 𝑝 7 distance scaling as a distance-7 surface code, with a prefactor 𝑁void = 256 versus ∼ 10 for the surface code—the cost of encoding 130 logical qubits at 29.0% rate versus 1 logical qubit at 1.0%. An adaptive scheduling protocol cuts the flag overhead to under 0.1% of measurement cycles at realistic noise levels.
The [[3L3, 2L3+2, 3]] CSS code on the Face-Centered Cubic (FCC) lattice achieves 67.7% encoding rate with weight-12 stabilizers, but its distance is capped at d = 3 by weight-3 logical operators sitting at tetrahedral interstitial voids. These logicals carry a geometric 􏰂ngerprint􏰀one error per triad sheet, all three converging at a known void􏰀that makes them detectable by a 􏰃ag ancilla measuring the three-body parity Zea Zeb Zec . The weight-3 logical 􏰃ips this parity from +1 to −1; the stabilizer syndrome stays silent, but the 􏰃ag sees it. Running R independent 􏰃ag rounds per syndrome cycle pushes the logical failure rate to p3+R, so the e􏰁ective distance is simply deff = 3 + R􏰀adjustable at runtime, with no changes to the code or its logical count. At R = 4 the FCC+􏰃ags code suppresses errors as well as a distance-7 surface code, while encoding 130 logical qubits at 29.0% rate versus 1 logical qubit at 1.0%. An adaptive scheduling protocol cuts the 􏰃ag overhead to under 0.1% of measurement cycles at realistic noise levels.
Abstract We present a CSS quantum error correcting code on the Face-Centered Cubic (FCC) lattice that combines surface-code-compatible K=4 connectivity with native inter-sheet logical gates between co-located CSS code blocks — a combination not previously achieved among well-studied codes. The construction has two parts. First, restricting the [[3L³, 2L³+2, 3]] FCC lattice code to a single triad sheet yields the sheet code with parameters [[L³, 2L, L]] at even L (or [[L³, L, L]] as a planar variant), uniform weight-4 stabilizers, and K=4 per-qubit connectivity. Three triad sheets share the FCC lattice geometry and can be deployed in one of three ways — monolithic 2D for small L, a three-layer stacked architecture for scalable deployment, or native 3D hardware — in each case encoding 6L logical qubits at distance L. Second, we introduce a fault-tolerant lattice surgery protocol that uses local FCC triangle measurements to implement joint Pauli measurements between logical qubits in different sheets. Each surgery operation adds O(L) two-qubit gates to the O(L³) per-round syndrome extraction cost. Under circuit-level depolarizing noise with MWPM decoding, the static memory threshold is p_th ≈ 0.63% and the surgery threshold is p_th ≈ 0.5%, both above the operating range of current quantum processors. Compared to surface code lattice surgery (also K=4, lattice surgery confined within a single patch), the sheet code provides 2L logicals per sheet at the same distance and connectivity; compared to bivariate bicycle codes (which target inter-block gates between physically separated modules at K=6 with long-range couplers), the sheet code achieves inter-sheet gates at K=4 with only short-range couplings between co-located sheets.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.