We construct a four-dimensional toric code on the D4 lattice, the densest lattice packing in four dimensions, by placing qubits on the 16L^4 triangles of its Delaunay decomposition into regular 16-cells. Each X-stabilizer acts on the 4 triangles of a tetrahedron, the smallest weight such a check can have, and each Z-stabilizer on the 8 triangles meeting along an edge. We prove that the code is [[16L^4, 6, 2L^2]] for every even L >= 4. A logical operator must cast a shadow covering a two-dimensional cross-section of the torus, and a single triangle shadows an area of at most 1/2, so at least 2L^2 qubits are needed; a close-packed plane of the lattice attains this bound exactly. Because the distance is twice the hypercubic value at equal side length, the code reaches distance d with 4d^2 physical qubits against 6d^2 for the hypercubic four-dimensional toric code. A hook-error analysis shows that the weight-8 checks cost no more at circuit level than the weight-4 ones. Neither sector has point-like excitations, and the code's equal-time slices are the face-centred-cubic lattice. Flat four-space has three regular honeycombs, and this code completes the regular case. All results are verified by published scripts.
We construct a four-dimensional toric code on the D4 lattice, the densest lattice packing in four dimensions, by placing qubits on the 16L^4 triangles of its Delaunay decomposition into regular 16-cells. Each X-stabilizer acts on the 4 triangles of a tetrahedron, the smallest weight such a check can have, and each Z-stabilizer on the 8 triangles meeting along an edge. We prove that the code is [[16L^4, 6, 2L^2]] for every even L >= 4. A logical operator must cast a shadow covering a two-dimensional cross-section of the torus, and a single triangle shadows an area of at most 1/2, so at least 2L^2 qubits are needed; a close-packed plane of the lattice attains this bound exactly. Because the distance is twice the hypercubic value at equal side length, the code reaches distance d with 4d^2 physical qubits against 6d^2 for the hypercubic four-dimensional toric code. A hook-error analysis shows that the weight-8 checks cost no more at circuit level than the weight-4 ones. Neither sector has point-like excitations, and the code's equal-time slices are the face-centred-cubic lattice. Flat four-space has three regular honeycombs, and this code completes the regular case. All results are verified by published scripts.
This work explores the composition of Microsoft's 4D geometric error-correcting codes — built on the 24-cell honeycomb cellulation — into higher-dimensional logical structures targeting 24D stability. While 4D codes have demonstrated 1,000-fold error rate reduction and single-shot correction on neutral atom arrays, this work asks how these codes compose when stacked or tensored into 8D, 12D, or 24D logical architectures. We investigate the stabilizer structure under composition, whether single-shot correctability survives dimensional lifting, and the mapping of composed code geometries onto neutral atom hardware with all-to-all connectivity. This bridges the gap between demonstrated 4D fault-tolerant codes and the theoretical framework for topological codes beyond dimension 2. Update, 30 July 2026 — constructive resolution. The distance question raised here is now settled by direct construction in 10.5281/zenodo.21698294, which builds the 4D toric code from its cubical chain complex and obtains [[96,6,4]] at L=2: k=6 exactly by GF(2) rank, and distance 4 proven exhaustively over every error pattern of weight ≤4. Distance 8 is therefore not obtainable from the 4D toric construction. That bounds the 4D toric family only and does not establish that no [[96,6,8]] code exists in another family — the quantum Singleton bound permits it comfortably at these parameters. An earlier wording of this note, added and corrected on 30 July 2026, overstated the scope. That work was carried out independently, from first principles, and reaches the same conclusion. Erratum, 2026-07-31 — the novelty claim is withdrawn, and what replaces it. The deposited file states that "nobody has published on composing 4D geometric codes into higher-dimensional logical structures" and that the bridge paper "has not been written." That is false. Composing chain-complex CSS codes is the homological (tensor) product of chain complexes, and the cellular chain complex of a product complex is the tensor product of the factors', so the composition of two 4D cubic toric codes is the 8D cubic toric code — a published family, and the subject of this record's own reference #9. This was checked, not assumed: an explicitly composed 8D code and an independently built 8D cubic toric code agree at all 8 grades on shape, nonzero count, both check-weight spectra and exact GF(2) rank. The composition constructs no new code. What replaces the claim is a quantitative answer to the question the record was actually reaching for. For the balanced hypercubic toric code at middle grade, n/k = L^D exactly and independently of D — verified in 8 of 8 computed cases, with the qubit count a combinatorial count and the logical count a measured pair of GF(2) ranks, up to n = 459,270 where both ranks were measured directly. Composing therefore changes how many logical qubits sit in one block and changes nothing else. Head to head at the same distance 81, the 8D composite [[459270, 70, d ≤ 81]] and the plain 4D code [[39366, 6, d ≤ 81]] both spend 6,561 physical qubits per logical qubit, to the digit — with the 8D version paying weight-10 stabilizers where the 4D pays weight-6. Distance does not rescue it either: by Künneth the distance of a product is bounded by the minimum over grade splittings, not the product of the factors' distances, so each added dimension is a chance to lose distance — the 5D product of [[486,6,9]] at grade 1 is [[1215, 5, 3]], proved exactly. On single-shot correction: there is nothing to lift. D = 4 at q = 2 already has metachecks in both sectors (ranks 80 and 80 for [[486,6,9]]), so 8D, 12D and 24D add no sector that was not already covered — and lifting can lose it, the 5D product at grade 1 having no X-sector metachecks at all (rank d₀ = 0). On hardware: all-to-all connectivity is real but was never the binding constraint. The smallest possible balanced 24D hypercubic toric code needs 4.5 × 10¹³ physical qubits, 1.5 × 10¹⁰ times the 3,000-atom array cited above. What stands: the homological product is the correct composition operation and the framework is well-posed; composition genuinely does raise dimension and does buy more logical qubits per block (6 → 70 from 4D to 8D); the metacheck question was the right one to ask; and the reference library is accurate. Two limits on the erratum itself: all computation uses the cubic 4-torus code [[486,6,9]], not Microsoft's 24-cell honeycomb code, whose parity-check matrices were never obtained — so nothing here is a statement about that code; and no exact distance is claimed at n ≥ 2430, the 8D figures being certified upper bounds with no lower bound proved. The 1,000-fold error-rate reduction and the neutral-atom hardware results were not tested and are neither confirmed nor refuted. Full computation, code and logs: 10.5281/zenodo.20808710 and branch writeup at /home/voodoo/branch_work/18779369_composition/RESULT.md. Note: the source file QEC_Neutral_Atom_24D_Research.md itself still contains the withdrawn claim verbatim (a published file cannot be edited in place) — this erratum is the authoritative correction until a new version replaces the file text.
This note corrects and improves the choice of code in the 4D cubic toric code family studied in Zenodo 20304690, 20301574, 20322507, and 20336622. At fixed k = 6 logical qubits (k = b2(T^4) = 6, a topological invariant), the torus shape is a free parameter. Minimizing physical-qubit count n = 6*prod(sides) at target distance d = minimum pairwise side product, the efficient frontier is the ISOTROPIC family (s,s,s,s) at n/d^2 = 6. The previously published anisotropic (2,4,4,4) -> [[768,6,8]] sits at n/d^2 = 12 (a 2x physical-qubit overhead) and is Pareto-dominated by the isotropic (3,3,3,3) -> [[486,6,9]]. [[486,6,9]] improves on [[768,6,8]] simultaneously on three axes:- 37% fewer physical qubits (486 vs 768);- higher distance (9 vs 8);- 2-6x lower logical failure rate under identical BP+OSD-CS-7 decoding and i.i.d. X noise across p = 0.04-0.08 (non-overlapping 95% confidence intervals, N = 4000 trials/rate). All parameters are verified by explicit construction (k by GF(2) rank; distance as minimum logical-operator weight, matching the geometric minimum non-contractible 2-surface). As a cross-check, the [[768,6,8]] failure rates measured here reproduce the previously published values to within statistical agreement, corroborating the original simulation. Method: direct code construction and Monte-Carlo decoder simulation (GF(2) linear algebra + ldpc 2.3.9 BP+OSD). No collapse/octonion engine is involved at any stage; every result is reproducible from the attached scripts (torus_code_4d.py, gate9_magnification.py, hh_threshold.py, burst_recheck_v2.py). Scope: results are for one decoder (BP+OSD-CS-7) and one noise model (i.i.d. X errors, perfect syndrome) -- the same setting as the prior deposits, so the comparison is direct. This is a code-parameter and decoder-performance result, not a hardware-viability claim; 4D toric codes require non-local qubit connectivity, which is not addressed here. Correction (2026-07-29): The Pareto-dominance claim in this record's title and abstract compares exactly two points — one isotropic (s,s,s,s) member against the single anisotropic [[768,6,8]] construction — and no frontier over the k=6 side-tuple space was computed, so dominance over the anisotropic family is not established by the evidence presented. What the comparison supports is the narrower statement that this isotropic code improves on [[768,6,8]] in n/d². A sweep of all k=6 side-tuples to n ≈ 2000 with the real n/d² frontier plotted is required before any Pareto claim can be restated. No erratum record exists for this item as of 2026-07-29; this is a self-correction pending that sweep. Addendum, 2026-07-31 — the efficient frontier, swept. The description above states that “the efficient frontier is the ISOTROPIC family (s,s,s,s) at n/d² = 6”. Sweeping every k=6 side-tuple with sides 2–6 and keeping the non-dominated points — using gate9_magnification.py, already attached to this record — shows that statement is too strong, and it is corrected here. Isotropic shapes uniquely attain the optimal ratio n/d² = 6, and every isotropic shape lies on the frontier. But they do not constitute the frontier: 4 of its 9 points are anisotropic. The reason is that isotropic shapes realise only square distances (4, 9, 16, 25, 36), so anisotropic shapes remain non-dominated wherever they supply an intermediate distance (6, 12, 20, 30) that no isotropic code of comparable size reaches. sidescoden/d²shape (2,2,2,2)[[96,6,4]]6.00isotropic (2,3,3,3)[[324,6,6]]9.00anisotropic (3,3,3,3)[[486,6,9]]6.00isotropic (3,4,4,4)[[1152,6,12]]8.00anisotropic (4,4,4,4)[[1536,6,16]]6.00isotropic (4,5,5,5)[[3000,6,20]]7.50anisotropic (5,5,5,5)[[3750,6,25]]6.00isotropic (5,6,6,6)[[6480,6,30]]7.20anisotropic (6,6,6,6)[[7776,6,36]]6.00isotropic Corrected statement. Isotropic tori uniquely minimise n/d², attaining 6 with equality; anisotropic tori are strictly worse on that ratio but still occupy the frontier at the distances isotropic shapes skip. The record's headline two-point comparison is unaffected — [[486,6,9]] has both fewer qubits than [[768,6,8]] (486 < 768) and greater distance (9 > 8), so that domination holds exactly as stated, as does the 37% qubit reduction. The sweep is bounded at side length 6 (n ≤ 7776). The alternating pattern continues beyond it by the same argument, but the frontier above n = 7776 was not enumerated.
We construct the 4D cubic toric code on a parameterized 4-torus T^4 = (Z/N0) x (Z/N1) x (Z/N2) x (Z/N3) with qubits on 2-cells, Z-stabilizers on 3-cells, and X-stabilizers on 1-cells, yielding [[6 * prod Ni, 6, min Ni*Nj]]. Three concrete sizes are benchmarked: [[96,6,4]] (cubic 2^4 torus), [[192,6,4]] (sides 2,2,2,4), and [[768,6,8]] (sides 2,4,4,4). Across 2,694 trials spanning three localized error families (weight-24 vertex bursts, weight-6 edge bursts, weight-{4,8,16} logical seeds) and three decoders (permissive GF(2) inversion, greedy minimum-syndrome, BP+OSD-CS-7 via the ldpc 2.3.9 package), we find: (1) all vertex and edge bursts at all three sizes are stabilizer-equivalent under the permissive decoder, with zero logical failures across 880 burst trials per decoder; (2) BP+OSD-CS-7 decodes every vertex burst to exact-zero residual at all three sizes (176/176 total) -- the decoder identifies the burst pattern itself as the minimum-weight correction; (3) logical seed errors are correctly classified into six distinct sectors, confirming the k=6 logical space is faithfully resolved. This empirically realizes the burst-protection consequence of the palindromic preservation theorem (Zenodo 10.5281/zenodo.20040776) at three independent code sizes, and closes the decoder gap left open by greedy methods at higher weights. A side finding: the cubic 2^4 torus yields distance d=4, not d=8 as sometimes quoted for the [[96,6,*]] toric code on the 24-cell; reaching d=8 requires sides (2,4,4,4) and n=768 qubits. This refines the parameter claim without affecting the palindromic theorem itself. The bundle contains the source (torus_code_4d.py, burst_recheck_v2.py), the full per-burst log (burst_recheck_all_sizes.log), parsed numeric results (results.json), and reproducibility instructions. Total compute: about 78 seconds on a single x86 VPS core for all three sizes. Entry 1 - Identifier: 10.5281/zenodo.20040776 - Relation: is supplement to - Resource type: Publication / Preprint Entry 2 - Identifier: 10.5281/zenodo.18815664 - Relation: is part of - Resource type: Publication / Other Erratum (2026-07-29): The 2,694 trials reported here at zero logical failures are a floor, not a failure curve — the record tested only burst weights 6 and 24, both inside the protected region and roughly 7 qubits below the boundary, so the failure boundary was never located. It has since been located on the isotropic [[486,6,9]]: localized bursts stay clean to weight 30 (0/20,000) and first fail at weight 31 (2/20,000), against weight 13 for unstructured random support, giving 2.5× structural tolerance at equal resolution. The curve is also strongly non-monotone, with protected spikes at complete balls — failure rate 0.802 at w=143, 0/5,000 at w=144 and w=145, back to 0.293 at w=146 — so no single-number burst-tolerance figure is meaningful for this code. See 10.5281/zenodo.21684236.
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.
The isotropic 4D cubic toric code (3,3,3,3) → [[486, 6, 9]] Pareto-dominates the previously published anisotropic (2,4,4,4) → [[768, 6, 8]]. The 4D cubic toric family previously studied at side configurations (2,4,4,4), (3,4,4,4), (4,4,4,4) is anisotropic — it fixes one torus direction at length 2. That choice is qubit-inefficient by a factor of two. The torus shape is a free parameter at fixed k = 6 logical qubits (k = b₂(T⁴) = 6 is topological, independent of cellulation). Optimising n = 6·∏(sides) at target distance d = min pairwise side product, the efficient frontier is the isotropic family (s,s,s,s) at n/d² = 6; the anisotropic (2,4,4,4) sits at n/d² = 12. Concretely: 486 vs 768 physical qubits (36.7% fewer), distance 9 vs 8. Under identical BP+OSD-CS-7 decoding and i.i.d. X noise the smaller code also achieves a lower logical failure rate at every physical error rate tested, with non-overlapping 95% confidence intervals. The improvement is simultaneous on all three axes: qubit count, code distance, and decoder performance. The corrected Pareto frontier is [[96,6,4]] → [[324,6,6]] → [[486,6,9]] → [[1152,6,12]] → [[1536,6,16]]. The previously published [[768,6,8]] is not on the frontier — it is strictly dominated. Scope limit, stated plainly: results are for BP+OSD-CS-7 under i.i.d. X noise with perfect syndrome extraction — the same setting as the prior deposits, so the comparison is like-for-like, but a full characterisation needs phenomenological and circuit-level noise and alternative decoders. Includes the head-to-head decoder script, raw results, curves, production data (.npz) and the distance verification code, so every figure is reproducible.
The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical CCCZ gate transversally.The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyperdiamond.This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical X and Z and only allows for the implementation of a transversal Clifford gate.This paper further develops the established connection between topological dimension and transversal gates in the Clifford hierarchy, generalizing the known designs for the implementation of transversal CZ and CCZ in two and three dimensions, respectively.