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.
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 NiNj]]. 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.
We measure the depolarizing-channel threshold of the 4D cubic toric code on the 4-torus T^4 under the BP + OSD-CS-7 decoder (ldpc 2.3.9, min-sum beta=0.625, 50 iterations, parallel schedule), at three sizes: [[768, 6, 8]] (sides 2,4,4,4), [[1152, 6, 12]] (sides 3,4,4,4), and [[1536, 6, 16]] (sides 4,4,4,4). For each (d, p) with p in {0.005, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08}, we sample 10,000 independent X-error vectors on the face qubits, decode the Z-syndrome with BP + OSD-CS-7, and classify the residual. Total: 270,000 trials. Across all three sizes and all nine rates the decoder cleared every syndrome (zero uncleared in 270,000 trials). Standard finite-size-scaling extraction places the BP + OSD threshold at p_th ≈ 0.10, 95% CI [0.09, 0.13] (consensus across four independent estimators: deep-below-threshold power-law fit, quadratic FSS restricted to the universal regime, quadratic FSS on all data, and three pairwise crossing-point extrapolations). This is roughly an order of magnitude above the surface-code MWPM threshold (~0.01) and is achieved without single-shot, repeated measurement, or correlated-decoder structure beyond what the off-the-shelf ldpc package provides. The empirical logical-failure curves cleanly separate by code distance up to p ≈ 0.06, with the larger codes suppressing failures by 50–500× at the matched physical rate. Mean correction weights match expected error weights to < 1% at every (d, p), confirming BP + OSD is finding genuine minimum-weight corrections rather than gaming the classifier. This is the stochastic-noise companion to the structured-burst result of DOI 10.5281/zenodo.20301574 on the same code family, which together establish that the 4D cubic toric family on T^4 is well below threshold across the entire near-term physical error-rate regime under a standard decoder. The bundle contains the source (torus_code_4d.py, burst_recheck_v2.py, noise_sweep_v2.py, noise_sweep_v3_high_p.py, fit_threshold_final.py), all per-rate raw counts as JSON (noise_sweep_v2_*.json and noise_sweep_v3_*.json), per-sweep stdout logs, the fit summary (threshold_fit_final.json), the threshold figure (threshold_curve_final.png), and the paper (THRESHOLD_PAPER.md). Reproducible in about 11 minutes on a single x86 VPS core.
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.
DOI 10.5281/zenodo.20301574 reports 2,694 trials at zero logical failures. That is a floor, not a curve — the failure boundary was never located. This note finds it on every axis, on the isotropic [[486,6,9]] rather than the superseded anisotropic [[768,6,8]]. The boundary axislargest clean burstfirst failure L1 ball radius (exhaustive, all 81 centres)r=2, weight 342 — 0/81r=3, weight 462 — 81/81 localized burst, exact weightw=30 — 0/20,000w=31 — 2/20,000 uniform random support (control)w=12 — 0/20,000w=13 — 1/20,000 Three findings the original could not have seen The original result was true and uninformative. All 81 weight-24 vertex bursts decode to exact zero residual, matching the record's 176/176 — but the record tested only weights 6 and 24, both inside the protected region and 7 qubits below the boundary. Structural tolerance quantified for the first time: 2.5x. At equal resolution, unstructured errors begin failing at w=13, geometrically localized bursts at w=30. Code distance is 9, so a distance-only guarantee covers only weight 4 — the remainder is degeneracy. The curve is strongly non-monotone, with protected spikes at complete balls. Failure rate reaches 0.802 at w=143, collapses to 0/5,000 at w=144 (=|B1|) and w=145, then returns to 0.293 at w=146. A wider protected window spans w≈320–352 around |B2|=342. No single-number burst-tolerance figure is meaningful for this code. The r=3 failure is intrinsic, not decoder-attributable The all-ones vector is a nontrivial logical operator with zero syndrome, and B3 ⊕ shell4 = all-ones with shell4 of weight 24 — so any maximum-likelihood decoder must prefer the weight-24 explanation. Failures reach all 63 non-trivial logical classes. Decoder: ldpc 2.4.1 BP+OSD-CS-7, min-sum, settings copied verbatim from the record's own burst_recheck_v2.py. 3,374,789 decodes. No zero is reported as "0%" anywhere — every zero carries Wilson and Clopper-Pearson upper bounds. Re-run at a different worker count (6→11): every count, rate, CI and seed byte-identical. Version 1.1 (2026-09-22): exact coset-weight audit Nothing in version 1.0 is withdrawn, and no count, rate or interval changes. This version adds an exact audit that changes how two of v1.0's findings should be read. Method. For a burst e with logical class l, the coset weight A(e) is the weight of the lightest error with the same syndrome and the same class. W(e) is the lightest with the same syndrome and any other class. Both are solved exactly as integer programs (scipy.optimize.milp / HiGHS), and every value quoted is a proven optimum. A minimum-weight (MW) decoder succeeds if and only if W > A. 1. The boundary is a coset-weight boundary. Random supports are not reduced at all (A = w). Localized bursts are reduced to about half their weight: the weight-24 vertex burst has A = 8, and w = 30 has A = 12–14. Every one of the 15 localized BP+OSD failures at w = 31–35 has W = A − 1, so an MW decoder fails on them too. Of the 11 random-support failures at w = 13–16, 7 have W = A − 1 and 4 are decoder shortfalls. Both families begin to fail at coset weight about 13–15. The 2.5× "structural tolerance" in v1.0 is real when counted in flipped qubits, but it is almost entirely the factor-of-two stabilizer reduction of a compact error. 2. Complete balls (A, W): B0 (8, 15); B1 (32, 33); B2 (6, 13); B3 (30, 24). The 342-qubit radius-2 ball is exactly equivalent to a weight-6 error with margin 7. The wide window around |B2| is genuine protection. 3. The radius-1 spike is a decoder near-tie, not protection built into the code. Throughout w = 143–146 the margin W − A is ±1. Over 40 sampled bursts per weight: wMW failuresBP+OSD failures1439/4031/401440/400/401450/400/401460/4010/40 At w = 146 every BP+OSD failure is a burst an MW decoder corrects. The v1.0 reading of the w = 144 drop as a protected spike of the code should be taken as a property of BP+OSD on a one-qubit near-tie. 4. Decoder strength. Re-decoding every boundary failure at OSD-CS order 12 instead of 7 rescued none, which is consistent with point 1. Files added: spike_exact.py / spike_exact2.py (MW audit; .log, .json), coset_weight_scan.py (A vs w; .log, .json), boundary_check.py (OSD-12 re-decode; .log), mechanism_check.py (syndrome and correction weights; .log), make_fig.py and fig_failure_curve.pdf, and VOODOO_SCREEN.md (an instrument screen only: a non-discriminative negative, and no result rests on it). All scripts run against the v1.0 code (code486.py, burst_sweep.py, same seeds). A manuscript based on this analysis is in preparation for Physical Review A.
Extension of the 4D cubic toric BP+OSD threshold study (Zenodo 20304690) to six lattices spanning four distance classes: - d=4 sub-family at 10,000 trials/rate: [[96,6,4]] (2,2,2,2), [[192,6,4]] (2,2,2,4), [[384,6,4]] (2,2,4,4) - d=8: [[768,6,8]] (2,4,4,4) at 10,000 trials/rate - d=12: [[1152,6,12]] (3,4,4,4) at 100,000 trials/rate - d=16: [[1536,6,16]] (4,4,4,4) at 100,000 trials/rate All under BP+OSD-CS-7 (ldpc 2.3.9, min-sum 0.625, 50 iterations, parallel decoding). i.i.d. X-error noise model, Z-syndrome decoding, 9 noise rates p ∈ {0.005, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08}. Three main findings: 1. Distance rule confirmed across the family. The minimum-pair-product rule d = min_{i<j} L_i · L_j predicts the realized distance in all six codes (6/6 match between prediction and decoder-confirmed distance). This is the family's distance parameterization. 2. Threshold is family-wide, not distance-class-specific. Pairwise crossings of the logical-failure-rate curves between any d=4 code and any d≥8 code cluster at p ≈ 0.08–0.13, the same range as d≥8 internal crossings. The d=4 sub-family does not introduce a separate threshold; it sits above the d≥8 codes at every p but crosses them at the same critical region. 3. Leading-order universal finite-size scaling fails at 100k precision. Four variants of the FSS ansatz log₁₀ P_L = c₀ + c₁ x + c₂ x² (+ c₃ x³) with x = (p − p_th) · d^(1/ν) were tried (orders 2 and 3; rate caps 0.05 and 0.02). All yield reduced χ² ∈ [8.5, 12.3], rejecting the model. The model-independent finite-size-crossing estimate p_th ≈ 0.11–0.13 sits above the FSS-fit central value p_th ≈ 0.089, indicating finite-d corrections to leading-order universal scaling that the simple ansatz does not capture. Companion to: Zenodo 20040776 (palindromic preservation methodology), 20301574 (empirical burst tolerance, three sizes), 20304690 (initial BP+OSD threshold for d ∈ {8, 12, 16}). Related identifiers: - 10.5281/zenodo.20304690 — IsSupplementTo (continues the threshold study with more codes and higher precision) - 10.5281/zenodo.20301574 — IsSupplementTo (uses the same three burst-tested sizes plus three new sizes) - 10.5281/zenodo.20040776 — IsSupplementTo (empirical companion to the methodology paper) - 10.5281/zenodo.18815664 — IsPartOf (24D library)
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.
We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical (d,d′)-Toric code with d′,d−d′≥2, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays.
Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results:
1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical (d,d′)-Toric complexes, resolving the local-existence question of Kubica and Preskill.
2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead.
3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate p below threshold, nd+1 noiseless sweep cycles recover the encoded state on a side-length-n(d,d′)-Toric code with probability at least 1−O(poly(n))pΩ(n), extending earlier guarantees for independent errors.