Restricting the [[3L3,2L3+2,3]] FCC lattice code to a single triad sheet gives the [[L3,2L,L]] sheet code, 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 logical interconnect: native fault-tolerant joint Pauli measurement between selected logical qubits in independently encoded, co-located code blocks, with no dedicated routing region. Its natural use is entanglement generation and parity measurement between such blocks, since the ZZ and XX merges characterized here are exactly the two measurements a logical Bell measurement requires. Every FCC triangle has one edge in each triad sheet, so products of triangle measurements implement joint Pauli measurements across sheets while the merged code retains d=L. Finite-size crossing estimates under circuit-level depolarizing noise at L∈{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 L2 physical qubits per logical, against 1.5L2 for a bussed toric array. That set is structured and limited: at the verified sizes of L∈{4,6,8}, each basis logical reaches L partners in one other sheet, giving 3L2 directly measurable pairs, corresponding to about 17% of all logical pairs, and the same pattern is conjectured for general even L values. Three limits are stated rather than deferred: only the joint measurements are fault-tolerant primitives, the composed three-sheet CNOT being verified but not yet fault-tolerantly characterized; all thresholds are finite-size crossings at three lattice sizes, not asymptotic values; and pairs outside the reachable set would need routing as in any other architecture.
Restricting the [[3L3,2L3+2,3]] FCC lattice code to a single triad sheet gives the [[L3,2L,L]] sheet code, 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 logical interconnect: native fault-tolerant joint Pauli measurement between selected logical qubits in independently encoded, co-located code blocks, with no dedicated routing region. Its natural use is entanglement generation and parity measurement between such blocks, since the ZZ and XX merges characterized here are exactly the two measurements a logical Bell measurement requires. Every FCC triangle has one edge in each triad sheet, so products of triangle measurements implement joint Pauli measurements across sheets while the merged code retains d=L. Finite-size crossing estimates under circuit-level depolarizing noise at L∈{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 L2 physical qubits per logical, against 1.5L2 for a bussed toric array. That set is structured and limited: at the verified sizes of L∈{4,6,8}, each basis logical reaches L partners in one other sheet, giving 3L2 directly measurable pairs, corresponding to about 17% of all logical pairs, and the same pattern is conjectured for general even L values. Three limits are stated rather than deferred: only the joint measurements are fault-tolerant primitives, the composed three-sheet CNOT being verified but not yet fault-tolerantly characterized; all thresholds are finite-size crossings at three lattice sizes, not asymptotic values; and pairs outside the reachable set would need routing as in any other architecture.
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.
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.
The three triad sheets of the face-centered cubic (FCC) lattice, at even lattice size L, are edge-disjoint but share their vertex and octahedral-void ancilla positions. Running the three sheet codes in a three-round time-multiplexed cycle on one chip needs 4L3 physical qubits against 6L3 for three separate blocks, a saving of exactly one third at every L, for instance, 864 qubits instead of 1296 at L=6. The schedule is implemented explicitly, with all three sheets on one shared ancilla set; comparing it against the effective single-sheet model used for the parameter sweeps shows agreement to within 5% at L=4 and 2.2% at L=6. The cost of the saving is that each sheet idles for two of every three sub-rounds, and the finite-size crossing estimate of the circuit-level threshold falls from 1.13±0.08% to 0.86±0.06%: a factor of 1.3, that is a 24% relative reduction, or 0.27 percentage points. Sweeping the idle-to-gate noise ratio r over 0,1,2,4 gives estimates decreasing monotonically from 1.13% to 0.71%. The dual X-basis memory gives 0.78±0.05%, overlapping the Z value. All values are finite-size crossings at L=4,6,8 from the full FCC circuit, not asymptotic thresholds, and the quoted spreads are dominated by finite-size drift rather than statistical error. Each sheet decomposes exactly into L independent rotated 2D toric codes (Proposition 1), so the memory is the toric code’s and no encoding-rate advantage is claimed; at d=4, the rate equals the rotated toric code’s. In the simulated noise model, where no channel couples two sheets, the sheets fail independently. Sub-threshold suppression survives the sharing at Λ(4→6)=12.3±1.1 and Λ(6→8)=5.6±1.2 at p=10−3, on a memory carrying 24 to 48 logical qubits. Block-level failure probabilities are reported alongside, since they grow with the logical count: at L=6 and p=10−3, the 36-logical chip loses at least one logical qubit in about one experiment in sixty. The L=6 instance, [[648,36,6]], needs 864 physical qubits, within the nominal atom count of current neutral-atom processors; no implementability claim is made beyond that count.
A fault-tolerant quantum computer needs a memory code and a source of magic states. On a neutral-atom array the two can use different codes. This paper connects a [[15,1,3]] color-code factory, which has a transversal T gate, to the FCC sheet code, a [[3L³, 6L, L]] CSS memory at K = 4 connectivity. The connection is a transfer: a logical state is moved from a color-code block into one sheet-code logical by one-bit teleportation. We build the transfer circuit with a flagged joint Z̄ₐZ̄ᵦ measurement and simulate it at L = 4 under circuit-level depolarizing noise. The noise-free truth table holds for all four basis states. The circuit distance is 4 in the Z sector and 3 in the X sector. At p = 10⁻³ the transfer fails at 0.17% (Z) and 1.7% (X); the sheet memory alone fails at 0.07% and below 0.27% over the same rounds. When all noise on the color-code block is turned off, the X-sector excess disappears. The cost of the transfer is therefore the distance-3 factory block failing while it waits, not the interface. Two further results support the architecture. The lattice rotation R: (x, y, z) ↦ (y, z, x) maps each of the three sheets onto the next. We verify at L = 4, 6, 8 that R is a CSS isomorphism and that it permutes the 2L logicals of a sheet. The sheets are copies of one code with a pairing fixed by the lattice, and the standard transversal CNOT between copies of a CSS code applies to any two sheets. One CNOT layer on R-paired qubits gives 2L logical CNOTs. Under noise, decoding the two sheets separately fails on the target sheet, with an apparent threshold near 0.3%, because the CNOT copies the control sheet's residual errors onto the target. A feed-forward decoder that propagates the decoded control errors through R restores suppression with L, with a crossing near 0.8% against the memory's 1.1%; with exact knowledge of the control errors the memory value is recovered. We also count the atoms used by the cross-sheet triangle-ribbon merge of the companion surgery paper: 3L/2 against 2L for a surface-code routing channel. We state what is not new. Each sheet is L rotated toric codes, so the memory is the toric code's. A transversal CNOT between copies of a CSS code is available to any stacked code, including surface-code patches on atom arrays. The sheet code adds a lattice-fixed pairing and the ribbon merge, not a throughput advantage; the pairing is not spatially local, since the distance between R-paired qubits grows as about 0.47L.
Quantum error correction is critical to the design and manufacture of scalable quantum computing systems. Recently, there has been growing interest in quantum low-density parity-check codes as a resource-efficient alternative to surface codes. Their adoption is hindered by the difficulty of compiling fault-tolerant logical operations. A key challenge is that logical qubits do not necessarily map to disjoint sets of physical qubits, which limits parallelism. We introduce clustered-cyclic codes, a quantum low-density parity-check code family with finite-size instances such as [[136,8,14]] and [[198,18,10]] that are competitive with state-of-the-art constructions. These codes admit a directly addressable logical basis, enabling highly parallel logical measurement layers. To leverage this structure, we propose parallel product surgery for quantum product codes. Using an auxiliary copy of the data patch and an engineered product-connection structure, the protocol performs many logical Pauli-product measurements in a single surgery round with small, fixed overhead. For clustered-cyclic codes, this yields surface-code-style maximal parallelism: up to k/2 disjoint Pauli-product measurements per round under explicit algebraic conditions. We prove that parallel product surgery preserves the code distance for hypergraph product codes and numerically verify distance preservation for the listed clustered-cyclic instances with k = 8. Finally, for the [[24,8,3]] clustered-cyclic code, treating half of the logical qubits as auxiliaries enables arbitrary parallel CNOTs on disjoint pairs; combined with symmetry-derived operations, these gates generate the full Clifford group fault-tolerantly.
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.