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.
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 [[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.
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.
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.
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.
Low-overhead quantum error-correction schemes are essential for enabling quantum computation on registers containing multiple logical qubits. For planar architectures with limited nearest-neighbor qubit connectivity, the surface code has emerged as the leading paradigm. Recent theoretical and experimental work has shown that a physical-qubit connectivity of degree three is sufficient to implement fault-tolerant quantum error correction. In this work, we study lattice surgery in the context of such trivalent architectures and introduce scalable circuit constructions to implement it. Compared with the four-valent measurement scheme, the trivalent lattice-surgery protocol reduces the required resources by O(d) qubits out of a total qubit count of O(d2) and by O(d) two-qubit gates out of a total two-qubit gate count of O(d3). We benchmark the logical fidelity of both lattice-surgery schemes in terms of experimentally realistic simulations targeting an implementation with a fluxonium qubit based architecture and find a potential improvement of up to ≈25% for distance-three. These results open a way for scalable planar trivalent qubit architectures to host a surface-code-based logical quantum processor.
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.