The Face-Centered Cubic (FCC) lattice supports two CSS quantum error correction codes: the full code [[3L^3, 2L^3+2, 3]] with K=12 connectivity, and the sheet code [[L^3, 2L, L]] obtained by restricting to one of three triad sheets. If the FCC lattice is identified with Planck-scale spacetime geometry, the Bekenstein-Hawking entropy bound S <= A/(4l_P^2) constrains the number of independent degrees of freedom (logical qubits) in any region. We show that the full FCC code violates this bound at every lattice size (k ~ L^3, volume scaling), while the sheet code satisfies it for L >= 4 (k ~ L, sub-area scaling). At L=4, the three-sheet code utilizes a fraction ln 2 ≈ 69% of the holographic capacity, the maximum at any admissible lattice size. The holographic principle thus acts as a code selection rule on the FCC lattice, uniquely selecting the sheet decomposition in which the 3D lattice reduces to 3L independent 2D toric codes as the physically admissible encoding.
The Face-Centered Cubic (FCC) lattice supports two CSS quantum error correction codes: the full code [[3L^3, 2L^3+2, 3]] with K=12 connectivity, and the sheet code [[L^3, 2L, L]] obtained by restricting to one of three triad sheets. If the FCC lattice is identified with Planck-scale spacetime geometry, the Bekenstein-Hawking entropy bound S <= A/(4l_P^2) constrains the number of independent degrees of freedom (logical qubits) in any region. We show that the full FCC code violates this bound at every lattice size (k ~ L^3, volume scaling), while the sheet code satisfies it for L >= 4 (k ~ L, sub-area scaling). At L=4, the three-sheet code utilizes a fraction ln 2 ≈ 69% of the holographic capacity, the maximum at any admissible lattice size. The holographic principle thus acts as a code selection rule on the FCC lattice, uniquely selecting the sheet decomposition in which the 3D lattice reduces to 3L independent 2D toric codes as the physically admissible encoding.
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.
Restricting the [[3L^3, 2L^3+2, 3]] FCC lattice code to a single triad sheet gives the sheet code [[L^3, 2L, L]], with uniform weight-4 stabilizers and K=4 active connectivity. This code is isomorphic to L disjoint rotated 2D toric codes, so its memory is the toric code's and no density advantage is claimed. The contribution is a primitive for joint Pauli measurement between logical qubits in distinct code blocks. Every FCC triangle has one edge in each triad sheet, so a weight-3 triangle measurement reaches one data qubit in each of three co-located blocks, and products of triangles implement joint Pauli measurements across sheets while the merged code retains d = L. Finite-size crossing estimates under circuit-level depolarizing noise at L in {4, 6, 8} are 1.07% +/- 0.05% for the ZZ-merge, 0.76% +/- 0.05% with planar boundaries, and 0.91% +/- 0.05% for the XX-merge. The value of the primitive is that, for the pairs it reaches, it needs no routing region. Surface-code surgery merges only adjacent patches, so joining arbitrary pairs costs a reservation of roughly 1.5 tiles per logical qubit; within the directly reachable set the sheet architecture avoids that reservation, using L^2 physical qubits per logical against 1.5L^2 for a bussed toric array. That set is structured and limited: each basis logical reaches L partners in one other sheet, giving 3L^2 directly measurable pairs, about 17% of all logical pairs. Three limits are stated rather than deferred. Only the joint measurements are fault-tolerant primitives, the composed three-sheet CNOT being verified at the truth-table level but not yet fault-tolerantly characterized. All thresholds are finite-size crossings at three lattice sizes, not asymptotic values. Pairs outside the reachable set, including all same-sheet pairs, would need routing as in any other architecture, and no advantage is claimed for them.
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 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.
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 three triad sheets of the face-centered cubic (FCC) lattice are edge-disjoint but share their vertex and octahedral-void ancilla positions. This paper quantifies what that sharing buys and what it costs. Running the three sheet codes in a three-round time-multiplexed cycle on one chip needs 4L^3 physical qubits, against 6L^3 for three separate blocks: a saving of exactly one third at every L, or 256 qubits instead of 384 at L = 4 and 864 instead of 1296 at L = 6. The price is measured here and is not small. Idle sheets accumulate depolarizing noise at rate 2p for two of every three rounds, and the circuit-level threshold falls from 1.13% +/- 0.08% to 0.86% +/- 0.06%, a factor of 1.3. Both figures are measured with the full FCC sheet-code circuit at L = 4, 6, 8, not with a surface-code proxy. Relation to the toric code. Each sheet decomposes exactly into L independent rotated 2D toric codes (Proposition 1), so the three-sheet code is 3L such codes sharing one ancilla set and one classical control pipeline. The sheets fail independently, and we say plainly that its memory behavior is the toric code's. We claim no encoding-rate advantage: at d = 4 the rate is 12.5%, which is identical to the rotated toric code [[16,2,4]], and the factor of two separating both from the rotated surface code is the usual periodic-boundary factor. Numerical results. Full circuit-level memory experiments in Stim with MWPM decoding, at L = 4, 6, 8 under the multiplex idle penalty. Sub-threshold suppression is Lambda(4->6) = 12.3 +/- 1.1 and Lambda(6->8) = 5.6 +/- 1.3 at p = 1e-3, on a memory carrying 24 to 48 logical qubits. These are the suppression factors of the underlying toric codes, confirmed to survive ancilla sharing and the idle penalty; we do not present them as evidence that this construction outperforms the surface code, and a matched-operating-point comparison is not made here. The L = 6 instance, [[648,36,6]] on 864 physical qubits, fits current neutral-atom processors.