The isotropic 4D cubic toric code on T^4 with sides (3,3,3,3) — the [[486,6,9]] code — has a transversal CNOT and a transversal Hadamard. Its non-Clifford completion had not previously been costed. This note re-verifies the code from an independent construction (n=486, rank(Hx)=rank(Hz)=240, k=6, logical weight 9), classifies its transversal gates, and derives the distillation cost. Two no-gos, both computed rather than assumed No transversal phase gate exists at all. Over all 2^486 subsets R of the qubits, the only product of S gates preserving the stabiliser group is the empty product. This is a complete classification, not a sample: the admissible subsets form a GF(2) nullspace, computed to have dimension 0. No transversal T. The matrix [Hx ; L_X] is not triorthogonal (3132 of 54285 pairs and 2268 of 5935160 triples have odd overlap), verified against a positive control on [[15,1,3]] which passes the identical test. Distillation cost The k=6 structure turns an apparent 6x qubit penalty into six 15-to-1 factories running in lockstep for the price of one, because every operation in the protocol is a block-transversal CNOT between distinct blocks. With single-shot error correction (a 4D-specific property), the space-time cost per output magic state is 7.7x lower than a rotated-surface-code factory at the same distance, despite using 7.04x more physical qubits. Output error coefficient is exactly 35 (the A_3 coefficient of the [15,11,3] Hamming code); the break-even fixed point is 0.141480. Code switching, quantified and rejected The only natural same-lattice partner has d=L, so matching d=9 costs n=26244 qubits for k=4 — 54x the qubits for fewer logical qubits. Distillation wins decisively. Relation to prior work This supersedes the branch note describing S and T as "unresolved": S was already resolved by |Y⟩-state injection in the 2026-06-23 session records. The open question was the cost, which this supplies. Two results here independently reproduce earlier corpus numbers from a different derivation (exact Hamming weight enumerator rather than 2^15 enumeration plus fit). Erratum (2026-07-29): The rail-parallelism design result in §5.2 — six 15-to-1 factories running in lockstep on the six rails of this code, driven by block-transversal CNOT, for a 7.67x space-time advantage — is prior art and is withdrawn as a novel claim. Xu et al., arXiv:2407.18490 (2024) §V.C.2, state that k magic states can be distilled in parallel in [[n,k,d]] patches, that this “mostly requires inter-block Clifford operations”, and explicitly that “the same protocol can be applied to distilling |T⟩ states in parallel using, e.g. a 15-to-1 distillation protocol, with the same scaling of the space and the time overhead”; their Appendix A additionally constructs the parallel magic-state injection that §7 of this record identified as its own unconstructed assumption. Separately, this record omits a cost the lockstep layout necessarily incurs: six rails sharing one block cannot post-select independently without rail-level addressability, so a round yields six magic states with probability p^6 or none, a space-time multiplier of 1/p^5 (1.08 at eps_in = 1e-3, 2.12 at 1e-2). The exact distillation results — coefficient 35, eps* = 0.141480, the depth-14 Koenig schedule — and both transversal no-gos are unaffected and are re-verified in the erratum. See 10.5281/zenodo.21697731.
The isotropic 4D cubic toric code on T^4 with sides (3,3,3,3) — the [[486,6,9]] code — has a transversal CNOT and a transversal Hadamard. Its non-Clifford completion had not previously been costed. This note re-verifies the code from an independent construction (n=486, rank(Hx)=rank(Hz)=240, k=6, logical weight 9), classifies its transversal gates, and derives the distillation cost. Two no-gos, both computed rather than assumed No transversal phase gate exists at all. Over all 2^486 subsets R of the qubits, the only product of S gates preserving the stabiliser group is the empty product. This is a complete classification, not a sample: the admissible subsets form a GF(2) nullspace, computed to have dimension 0. No transversal T. The matrix [Hx ; L_X] is not triorthogonal (3132 of 54285 pairs and 2268 of 5935160 triples have odd overlap), verified against a positive control on [[15,1,3]] which passes the identical test. Distillation cost The k=6 structure turns an apparent 6x qubit penalty into six 15-to-1 factories running in lockstep for the price of one, because every operation in the protocol is a block-transversal CNOT between distinct blocks. With single-shot error correction (a 4D-specific property), the space-time cost per output magic state is 7.7x lower than a rotated-surface-code factory at the same distance, despite using 7.04x more physical qubits. Output error coefficient is exactly 35 (the A_3 coefficient of the [15,11,3] Hamming code); the break-even fixed point is 0.141480. Code switching, quantified and rejected The only natural same-lattice partner has d=L, so matching d=9 costs n=26244 qubits for k=4 — 54x the qubits for fewer logical qubits. Distillation wins decisively. Relation to prior work This supersedes the branch note describing S and T as "unresolved": S was already resolved by |Y⟩-state injection in the 2026-06-23 session records. The open question was the cost, which this supplies. Two results here independently reproduce earlier corpus numbers from a different derivation (exact Hamming weight enumerator rather than 2^15 enumeration plus fit). Erratum (2026-07-29): The rail-parallelism design result in §5.2 — six 15-to-1 factories running in lockstep on the six rails of this code, driven by block-transversal CNOT, for a 7.67x space-time advantage — is prior art and is withdrawn as a novel claim. Xu et al., arXiv:2407.18490 (2024) §V.C.2, state that k magic states can be distilled in parallel in [[n,k,d]] patches, that this “mostly requires inter-block Clifford operations”, and explicitly that “the same protocol can be applied to distilling |T⟩ states in parallel using, e.g. a 15-to-1 distillation protocol, with the same scaling of the space and the time overhead”; their Appendix A additionally constructs the parallel magic-state injection that §7 of this record identified as its own unconstructed assumption. Separately, this record omits a cost the lockstep layout necessarily incurs: six rails sharing one block cannot post-select independently without rail-level addressability, so a round yields six magic states with probability p^6 or none, a space-time multiplier of 1/p^5 (1.08 at eps_in = 1e-3, 2.12 at 1e-2). The exact distillation results — coefficient 35, eps* = 0.141480, the depth-14 Koenig schedule — and both transversal no-gos are unaffected and are re-verified in the erratum. See 10.5281/zenodo.21697731.
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.
SUPERSEDED DUPLICATE — do not cite this record. Cite 10.5281/zenodo.21698294 instead. Both records were deposited on 2026-07-30 and contain the same paper. This one carries an earlier, superseded NOTE.md whose resource-scaling section still claimed L≈6–7 (d=36–49); the surviving record carries the corrected text (measured L=4, [[1536,6,16]]) and the toric4d_lowp.py stratified estimator behind it. The independent cross-validation scripts that were unique to this record (magic_state4-7.py) have been added to the surviving record. Flagged 2026-07-31. We simulate a 15-to-1 magic-state factory under full circuit-level depolarizing noise and ask whether a 4D toric code — which corrects errors single-shot, without repeated syndrome rounds — relieves the factory's dominant cost. It does not. Distillation output error is confirmed at 35p³ by exact enumeration and by stim, but the factory's circuit-level error floor is first order in the gate error rate and schedule-dependent, measured at 12–15·p_gate for the real [[15,1,3]] quantum Reed–Muller code with all 14 stabilizers. This is 3.5× larger than a previously reported 4.18·p_gate, which we trace to a circuit that omitted the code's 10 Z-type checks; that figure is withdrawn here. The floor originates in the final syndrome sweep: faults deposited on data qubits by the last round of gates are never checked by anything after them. Whichever check type is measured last, that sector's faults escape, so rescheduling shifts the floor between sectors and interleaving shaves roughly 20%, but cannot remove it. Reordering, redundant checks and flag qubits leave it first order; decoding weight-1 errors instead of discarding them makes it 5× worse. We construct the 4D toric code on an L⁴ torus from its cubical chain complex and verify [[486,6,9]] at L=3 (k=6 exactly, explicit weight-9 logicals in both sectors, d=4 exact at L=2 by exhaustive search), and measure a code-capacity threshold between 5% and 7% under BP+OSD. Single-shot repair using the 80 independent metachecks recovers near-perfect-syndrome performance, but only while syndrome noise stays roughly 10–30× below the data error rate: it improves logical error by 195× at q=10⁻³ and by only 4.5× at q=10⁻². Crucially, metachecks constrain the syndrome, not the data, and cannot see faults deposited directly on data qubits by the final sweep — which is precisely where the factory floor originates. Single-shot therefore buys latency, not factory fidelity. Because the factory's Cliffords act on encoded blocks, p_gate is the logical error rate of the hosting code. The correction moves the surface-code requirement from d=15 to d=17 (578 physical qubits per patch); on the 4D toric side, the smallest sufficient block is L=4, [[1536,6,16]], with [[486,6,9]] missing the fidelity demand by four to five orders of magnitude. We situate the result against Xu et al. (arXiv:2407.18490), whose parallel distillation across qLDPC patches reduces space–time cost but leaves the per-factory floor untouched. A corrections log records eight claims withdrawn or corrected during this work, including our own earlier 4.18·p_gate figure and a d=15 requirement that rested on a single error event. Section 6 states what is not established: these are code-capacity numbers for the 4D code, the distance requirement is quoted at p=3–5% rather than 10⁻³, d=L² is verified exactly only at L=2, the Λ extrapolations are long, and no end-to-end injection simulation was run. Correction, 2026-07-31. The files deposited with this record on 2026-07-30 predate a low-p re-measurement of the resource-scaling section. The originally deposited text stated that hosting the factory requires L≈6–7 (d=36–49, n≈7800–14400); that figure came from fitting a single suppression factor Λ to near-threshold points at p=3–5%. The single-Λ ansatz does not hold for this code: the measured suppression per two units of distance runs from 10.9 (d=4→9 at p=0.02) to 793 (d=9→16 at p=0.01). Re-measuring at the factory's actual operating regime with a weight-stratified estimator gives d ≥ 14.5 at p=0.02 and d ≥ 9.7 at p=0.01, so the smallest sufficient 4D block is L=4, [[1536,6,16]], not L=6–7. The L≈6–7 figure is withdrawn. These are point estimates whose upper confidence bands remain wide at the smallest p, and should be read as order-of-magnitude. The corrected text and the estimator script (toric4d_lowp.py) are not in this record's file set; replacing files requires a new version, which will carry the corrected note. Nothing in Sections 1–4 — the 35p³ distillation result, the 12–15·p_gate factory floor, the last-sweep mechanism, or the d=17 surface-code requirement — is affected.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]] where β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S, SHS=X, and CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
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.
We propose a family of error-detecting stabilizer codes with an encoding rate of 1/3 that permit a transversal implementation of the gate $T=\mathrm{exp}(\ensuremath{-}i\ensuremath{\pi}Z/8)$ on all logical qubits. These codes are used to construct protocols for distilling high-quality ``magic'' states $T\left|+\right\ensuremath{\rangle}$ by Clifford group gates and Pauli measurements. The distillation overhead scales as $O({\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon}))$, where $\ensuremath{\epsilon}$ is the output accuracy and $\ensuremath{\gamma}={\mathrm{log}}_{2}(3)\ensuremath{\approx}1.6$. To construct the desired family of codes, we introduce the notion of a triorthogonal matrix, a binary matrix in which any pair and any triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal T gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating triorthogonal matrices is proposed. Our techniques lead to a twofold overhead reduction for distilling magic states with accuracy $\ensuremath{\epsilon}\ensuremath{\sim}{10}^{\ensuremath{-}12}$ compared with previously known protocols.
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.