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.
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 ngerprintone error per triad sheet, all three converging at a known voidthat 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 eective distance is simply deff = 3 + Radjustable 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.
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 (HXHZT=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. The minimum distance d=3 is proven exactly by exhaustive elimination of all weight-≤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 3L3 edges but only L3−2 independent stabilizer constraints, leaving k=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=0.001 (and 63x at p=0.0005), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.
Abstract We provide a circuit-level numerical validation of the three-sheet time-multiplexed FCC sheet code, a quantum low-density parity-check (qLDPC) construction with parameters [[3L 3 , 6L, L]] derived from the three triad sheets of the face-centered cubic lattice. At L = 4 the code packs 24 logical qubits into 192 data qubits at distance 4, an encoding rate of 12.5%, higher than any other published code at this distance. Three independent sheet-codes share one chip; ancillas are shared, stabilizers are extracted in three sequential rounds, and the inactive sheets accumulate depolarizing noise during each round at rate 2p. We simulate the full circuit-level memory experiment with explicit CNOT cascades, depolarizing noise on every gate, and minimum-weight perfect matching decoding via PyMatching on Stim's detector error model. Threshold under the multiplex idle penalty falls between 0.4% and 0.5%, in agreement with the original construction paper. Distance scaling is sharp: the per-logical error rate suppresses by Λ(4→6) = 11.2 between consecutive even L at p = 0.001, and Λ(6→8) = 7.8. At L = 8 the [[1536, 48, 8]] code achieves a per-logical error rate of 5.9 × 10⁻⁵, more than an order of magnitude below the physical rate, on a 48-logical-qubit memory. These suppression factors, measured in simulation deeply sub-threshold, are over five times those recently reported on noisy hardware on a single-logical surface-code memory; an apples-to-apples comparison at matched operating point requires either future hardware experiments on this code or a surface-code simulation at the same deeply sub-threshold operating point and is not made here. The natural hardware target is the L = 6 instance: [[648, 36, 6]] encodes 36 logical qubits at distance 6 in 864 physical qubits, fitting Atom Computing's 1180-qubit Phoenix processor and exceeding every prior multi-logical hardware demonstration in distance and logical-qubit count. The L = 4 instance overlaps with Atom Computing and Microsoft's 24-logical [[4, 2, 2]] demonstration, providing a calibration point at quadrupled distance (d = 4 vs d = 2) and full error correction in place of error detection.
The [[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 4L3 triangular faces of the lattice. The Z-type ones are the 4L3 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 2L3−1 independent logical classes. We prove this count analytically from the homology of the 3-torus, not only numerically: the cycle space has dimension 25L3+1 , the boundary space has dimension 25L3−2 , and subtracting the stabilizer rank leaves 2L3−1 for every even L. So k=(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 ZeaZebZec at a face anticommutes with four octahedral stabilizers, so it is not measurable, and no measurement of any kind can distinguish ∣ψ⟩ from L^T∣ψ⟩ for arbitrary encoded ∣ψ⟩ . Removing the sector requires adjoining all 2L3−1 independent directions, which returns k=3 . Verified exactly over GF(2) at L=4,6,8,10 .
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.