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Modular Assembly of High-Performance Logical Blocks from the Lorentzian Causal Diamond: Pareto-Optimal Finite-Block Codes, Asymmetric Distance Families, and an E₈ Structural Obstruction

Yannick Schmitt

Published Apr 10, 2026• 0 citations
Publisher

Abstract

This repository contains the complete research artifact and computational verification suite for scaling the discrete Lorentzian causal diamond into high-performance Quantum Low-Density Parity-Check (qLDPC) and finite-block codes. By systematically exploring algebraic amplification (Hypergraph Product), geometric tessellation (toric scaling), and lattice enrichment ($E_8$ Lorentzian substrates), this work identifies precise structural obstructions in existing approaches and introduces novel, Pareto-optimal CSS codes tailored for near-term hardware in the $N \sim 100\text{--}200$ physical qubit regime. Scientific Contributions The Augmented-Seed Code Family: The central discovery that a specific weight-3 row (one of 64 in two $B_4$ symmetry orbits) appended to the primal causal diamond seed raises the classical distance from 4 to 6. This breaks the intrinsic distance ceiling of the basic geometry, unlocking a family of finite-block codes that significantly outperform planar codes (e.g., surface/toric codes) at small block sizes. Methodological Warning on Sampling Blindness: Demonstration that Belief Propagation (BP)-style random sampling fails catastrophically in high-dimensional kernels. Earlier sampled bounds of $d \ge 67$ for the `[[193, 25]]` code are formally retracted; an exact Integer Linear Program (ILP) oracle is introduced, definitively proving $d = 4$. The $E_8$ Structural Obstruction (Theorem 4.5): Rigorous proof that the 28-qubit $E_8$ Lorentzian code cannot be punctured to yield a valid quantum code. Its 7-disconnected-4-cycle geometry guarantees $d_Z = 1$ after any single row drop, pointing toward the Euclidean $E_8$ root lattice as the necessary next step. Naïve CSS Lift Theorem (Theorem 5.3): Algebraic proof that setting $H_X = \ker(H_Z)$ definitionally forces $k=0$, regardless of chain or torus topology, resolving anomalies in earlier geometric scaling attempts. Highlighted Code FamiliesThe augmented geometric construction yields a frontier of codes optimized for neutral-atom and trapped-ion architectures, including: [[112, 4, (6, 6)]]: Achieves an exceptional figure of merit $kd^2/N \approx 1.29$, making it a highly efficient logical block for near-term implementation. [[176, 32, (3, 6)]: Features an asymmetric distance profile naturally aligned with the $Z$-biased noise characteristic of superconducting hardware. [[208, 16, 6]]: A self-hypergraph product code with $d=6$ proven algebraically via the Tillich–Zémor theorem. Artifact Contents Verification Suite (`verification_modular_assembly.py`): A unified, self-contained Python script using `scipy.optimize.milp` that computationally verifies every theorem, constructs all augmented seeds, performs exact distance certifications via ILP in under 100ms per operator, and reproduces the geometric obstructions. Manuscript: Complete LaTeX source files and compiled PDF of the paper.

DOI
10.5281/zenodo.19484042

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A Lorentzian CSS Duality in Causal Diamond Quantum Error-Correcting Codes - Four Codes from One Geometry via Orientation Reversal

Yannick Schmitt

Abstract This paper shows that the discrete Lorentzian causal diamond D — the two-complex built on the twelve lightlike nearest-neighbour vectors of the ternary Minkowski lattice {-1, 0, +1}^4 — generates not two but four CSS quantum error-correcting codes via a geometric duality, and that this duality makes the distance asymmetry dZ = 2 of the original construction algebraically inevitable rather than merely observed. The central mechanism is that the causal diamond supports two natural orientations of its boundary: the Lorentzian orientation, in which past links are incoming, produces the temporal charge n^0_eff = 12 and motivates an all-ones X-check; the Euclidean orientation, in which all links are outgoing and the boundary sum vanishes, yields a dual code family in which the 21 plaquettes serve as X-type checks and the three spatial-axis groups serve as Z-type checks. These two orientations are related by Wick rotation t → ix, so the duality between the primal and dual code families is a quantum error-correction realisation of Wick rotation. The four codes derived from this geometry are: Code Parameters Highlights Code I [[12, 4, (4,2)]] Rate 1/3; corrects X-errors, detects Z-errors Code II [[12, 1, (4,3)]] Balanced; circuit-level threshold p_c ≈ 3.5% Dual A [[12, 2, (2,6)]] New; corrects all weight-1 and weight-2 Z-errors Dual II [[12, 1, (3,4)]] New; dZ = 4, preferred for dephasing-dominated hardware Additional contributions include: a two-stage combined protocol that measures the 21 plaquettes alternately in Z- and X-basis to correct both error types simultaneously (P_log = 0.006 at p = 0.01 with k_eff = 2), a Pigeonhole No-Go theorem proving that dZ ≥ 3 with k ≥ 2 is impossible in the primal CSS family, and an X-Decoration Equivalence theorem extending this bound to weight-≤6 non-CSS codes. Repository Contents Code: Numerical verification script Paper: Full PDF pre-print and original LaTeX source files. License Information Please note the dual-licensing structure of this repository: Software/Source Code: Licensed under the Apache License 2.0. PDF Document & LaTeX Source: Licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0).

•Mar 31, 2026•0 citations

Logical computation with canonical lifted product codes

Han Zheng, Guo Zheng, Liang Jiang +1 more

High-rate quantum low-density parity-check (qLDPC) codes encode many logical qubits with low physical-qubit overhead, but realizing efficient fault-tolerant computation on such dense encodings remains a major challenge. Generic, code-agnostic techniques such as code surgery and gate teleportation apply broadly, but are difficult to make modular, low-overhead, and fully certifiable on complex high-rate codes whose structure is left unexploited. Here we overcome these obstacles by co-designing the code together with its logical instruction set for a broad family of canonical lifted-product (LP) codes with cyclic symmetry. We show that these codes admit a canonical logical basis, in which conjugate logical operators are organized into rows and columns of cyclic orbits inherited directly from the underlying classical codes, analogous to the structure that makes hypergraph-product codes so tractable. This canonical basis enables universal logical processing using magic-state input and a constant number of reusable seed surgery gadgets---only two (resp. four) for a $[[1122,148,\leq 20]]$ (resp. $[[4350,1224,\leq 20]]$) LP code. A fixed extractor smaller than half the data block also enables arbitrarily high-weight logical Pauli-product measurements (PPMs) while preserving the code's cyclic symmetry. These codes also enable rigorously fault-tolerant parallel logic, including PPMs across all logical qubits and magic-state injection from many surface codes into a full LP block. For example, a $[[468,36,20]]$ (resp. $[[952,112,17]]$) code supports simultaneous measurements of arbitrary logically disjoint Pauli products with an ancilla overhead of roughly three (resp. four) times the data block size. These results advance the frontier of fault-tolerant quantum computation on ultra-high-rate quantum architectures.

quant-pharXiv:2607.28605•Jul 31, 2026•0 citations

Composing 4D Geometric Codes into Higher-Dimensional Logical Structures: A Framework for 24D Quantum Error Correction Stability

James Johnstone Jardine

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.

•Feb 26, 2026•1 citation

Continuous Relaxation Decoding for CSS Quantum LDPC Codes: Energy Landscape, Basin Geometry, Gradient Dynamics, and Hardware Implementation

Yannick Schmitt

This repository contains the full research for a novel approach to decoding Quantum Low-Density Parity-Check (QLDPC) codes via continuous optimization. Moving away from traditional message-passing or combinatorial algorithms, this work introduces a syndrome-agnostic energy landscape—a "squared-spring" potential where global minima correspond exactly to valid codewords. The decoder operates as a codeword projector, using gradient dynamics to navigate the continuous relaxation of the binary error space. Key Contributions The Energy Landscape: Formalization of the squared-spring Hamiltonian $H(v)$, proving that valid codewords are the only points with zero energy and zero gradient (Theorem 3.4). Basin Geometry & Convergence: Analytical derivation of the exact Hessian and gradient at valid codewords (Theorems 4.1, 4.3). The work defines three nested radii ($r_{loc}$, $r_{barrier}$, $r_{emp}$) that characterize the decoder’s convergence behavior. The Girth–Trap Hypothesis: An empirical finding that challenges existing intuition; data demonstrates that higher girth does not necessarily prevent decoder trapping, with girth-8 codes observed to trap at higher rates than girth-4 variants. Optimal λ Heuristic: Discovery of a universal scaling law $\lambda^* \approx (3/16)\bar{d}$ for the regularization parameter, verified across eight independent CSS code families. Hardware Acceleration: A high-throughput implementation leveraging PyTorch and CUDA, achieving >86,000 syndromes per second, representing a ~28,000× speedup over serial Python implementations. Contents Core Library: `continuous_relaxation_decoder.py` — A modular Python library for CSS code construction, GF(2) linear algebra, and energy-based decoding. Verification Suite: `verification_ContinuousRelaxationDecoder.py` — A self-contained script that numerically verifies every theorem and proposition in the paper to machine precision. Computational Experiments: Momentum and Langevin dynamics comparisons. Failure mode census and pseudo-codeword barrier height calculations. Causal Diamond (CD) geometry analysis and [[208, 16, 6]] distance certification. Manuscript: LaTeX source and compiled PDF of the paper. Testbed CodesThe results are benchmarked on several high-performance QLDPC codes, including: [[193, 25, 4]] D4 Causal Diamond Hypergraph Product (HGP) code. [[112, 4, (6,6)]] and [[176, 32, (3,6)]] augmented-seed codes. Various Generalised Bicycle and CSS-LDPC code families.

•Apr 10, 2026•0 citations

Planar Quantum Low-Density Parity-Check Codes with Open Boundaries

Zijian Liang, Jens Niklas Eberhardt, Yu-An Chen

Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions 6 ≤ k ≤ 13 , e.g., ⟦ 78 , 6 , 6 ⟧ , ⟦ 107 , 7 , 7 ⟧ , ⟦ 268 , 8 , 12 ⟧ , ⟦ 405 , 9 , 15 ⟧ , ⟦ 348 , 10 , 13 ⟧ , ⟦ 450 , 11 , 15 ⟧ , ⟦ 386 , 12 , 12 ⟧ , and ⟦ 362 , 13 , 11 ⟧ , all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a ⟦ 282 , 12 , 14 ⟧ code, exhibiting an efficiency metric ( k d 2 / n ) an order of magnitude higher than the surface code. The construction combines boundary-anyon condensation with the “lattice-grafting” optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators the distance of which scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.

arXiv:2504.08887•PRX Quantum•Nov 12, 2025•4 citations

QGPU: Parallel logic in quantum LDPC codes

Boren Gu, Andy Zeyi Liu, Armanda O. Quintavalle +3 more

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.

quant-pharXiv:2603.05398•Mar 6, 2026•0 citations
This repository contains the complete research artifact and computational verification suite for scaling the discrete Lorentzian causal diamond into high-performance Quantum Low-Density Parity-Check (qLDPC) and finite-block codes. By systematically exploring algebraic amplification (Hypergraph Product), geometric tessellation (toric scaling), and lattice enrichment (E8E_8E8​ Lorentzian substrates), this work identifies precise structural obstructions in existing approaches and introduces novel, Pareto-optimal CSS codes tailored for near-term hardware in the N∼100–200N \sim 100\text{--}200N∼100–200 physical qubit regime. Scientific Contributions The Augmented-Seed Code Family: The central discovery that a specific weight-3 row (one of 64 in two B4B_4B4​ symmetry orbits) appended to the primal causal diamond seed raises the classical distance from 4 to 6. This breaks the intrinsic distance ceiling of the basic geometry, unlocking a family of finite-block codes that significantly outperform planar codes (e.g., surface/toric codes) at small block sizes. Methodological Warning on Sampling Blindness: Demonstration that Belief Propagation (BP)-style random sampling fails catastrophically in high-dimensional kernels. Earlier sampled bounds of d≥67d \ge 67d≥67 for the [[193, 25]] code are formally retracted; an exact Integer Linear Program (ILP) oracle is introduced, definitively proving d=4d = 4d=4. The E8E_8E8​ Structural Obstruction (Theorem 4.5): Rigorous proof that the 28-qubit E8E_8E8​ Lorentzian code cannot be punctured to yield a valid quantum code. Its 7-disconnected-4-cycle geometry guarantees dZ=1d_Z = 1dZ​=1 after any single row drop, pointing toward the Euclidean E8E_8E8​ root lattice as the necessary next step. Naïve CSS Lift Theorem (Theorem 5.3): Algebraic proof that setting HX=ker⁡(HZ)H_X = \ker(H_Z)HX​=ker(HZ​) definitionally forces k=0k=0k=0, regardless of chain or torus topology, resolving anomalies in earlier geometric scaling attempts. Highlighted Code FamiliesThe augmented geometric construction yields a frontier of codes optimized for neutral-atom and trapped-ion architectures, including: [[112, 4, (6, 6)]]: Achieves an exceptional figure of merit kd2/N≈1.29kd^2/N \approx 1.29kd2/N≈1.29, making it a highly efficient logical block for near-term implementation. [[176, 32, (3, 6)]: Features an asymmetric distance profile naturally aligned with the ZZZ-biased noise characteristic of superconducting hardware. [[208, 16, 6]]: A self-hypergraph product code with d=6d=6d=6 proven algebraically via the Tillich–Zémor theorem. Artifact Contents Verification Suite (verification_modular_assembly.py): A unified, self-contained Python script using scipy.optimize.milp that computationally verifies every theorem, constructs all augmented seeds, performs exact distance certifications via ILP in under 100ms per operator, and reproduces the geometric obstructions. Manuscript: Complete LaTeX source files and compiled PDF of the paper.
Abstract This paper shows that the discrete Lorentzian causal diamond D — the two-complex built on the twelve lightlike nearest-neighbour vectors of the ternary Minkowski lattice {-1, 0, +1}^4 — generates not two but four CSS quantum error-correcting codes via a geometric duality, and that this duality makes the distance asymmetry dZ = 2 of the original construction algebraically inevitable rather than merely observed. The central mechanism is that the causal diamond supports two natural orientations of its boundary: the Lorentzian orientation, in which past links are incoming, produces the temporal charge n^0_eff = 12 and motivates an all-ones X-check; the Euclidean orientation, in which all links are outgoing and the boundary sum vanishes, yields a dual code family in which the 21 plaquettes serve as X-type checks and the three spatial-axis groups serve as Z-type checks. These two orientations are related by Wick rotation t → ix, so the duality between the primal and dual code families is a quantum error-correction realisation of Wick rotation. The four codes derived from this geometry are: Code Parameters Highlights Code I [[12, 4, (4,2)]] Rate 1/3; corrects X-errors, detects Z-errors Code II [[12, 1, (4,3)]] Balanced; circuit-level threshold p_c ≈ 3.5% Dual A [[12, 2, (2,6)]] New; corrects all weight-1 and weight-2 Z-errors Dual II [[12, 1, (3,4)]] New; dZ = 4, preferred for dephasing-dominated hardware Additional contributions include: a two-stage combined protocol that measures the 21 plaquettes alternately in Z- and X-basis to correct both error types simultaneously (P_log = 0.006 at p = 0.01 with k_eff = 2), a Pigeonhole No-Go theorem proving that dZ ≥ 3 with k ≥ 2 is impossible in the primal CSS family, and an X-Decoration Equivalence theorem extending this bound to weight-≤6 non-CSS codes. Repository Contents Code: Numerical verification script Paper: Full PDF pre-print and original LaTeX source files. License Information Please note the dual-licensing structure of this repository: Software/Source Code: Licensed under the Apache License 2.0. PDF Document & LaTeX Source: Licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0).
High-rate quantum low-density parity-check (qLDPC) codes encode many logical qubits with low physical-qubit overhead, but realizing efficient fault-tolerant computation on such dense encodings remains a major challenge. Generic, code-agnostic techniques such as code surgery and gate teleportation apply broadly, but are difficult to make modular, low-overhead, and fully certifiable on complex high-rate codes whose structure is left unexploited. Here we overcome these obstacles by co-designing the code together with its logical instruction set for a broad family of canonical lifted-product (LP) codes with cyclic symmetry. We show that these codes admit a canonical logical basis, in which conjugate logical operators are organized into rows and columns of cyclic orbits inherited directly from the underlying classical codes, analogous to the structure that makes hypergraph-product codes so tractable. This canonical basis enables universal logical processing using magic-state input and a constant number of reusable seed surgery gadgets---only two (resp. four) for a [[1122,148,≤20]][[1122,148,\leq 20]][[1122,148,≤20]] (resp. [[4350,1224,≤20]][[4350,1224,\leq 20]][[4350,1224,≤20]]) LP code. A fixed extractor smaller than half the data block also enables arbitrarily high-weight logical Pauli-product measurements (PPMs) while preserving the code's cyclic symmetry. These codes also enable rigorously fault-tolerant parallel logic, including PPMs across all logical qubits and magic-state injection from many surface codes into a full LP block. For example, a [[468,36,20]][[468,36,20]][[468,36,20]] (resp. [[952,112,17]][[952,112,17]][[952,112,17]]) code supports simultaneous measurements of arbitrary logically disjoint Pauli products with an ancilla overhead of roughly three (resp. four) times the data block size. These results advance the frontier of fault-tolerant quantum computation on ultra-high-rate quantum architectures.
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 repository contains the full research for a novel approach to decoding Quantum Low-Density Parity-Check (QLDPC) codes via continuous optimization. Moving away from traditional message-passing or combinatorial algorithms, this work introduces a syndrome-agnostic energy landscape—a "squared-spring" potential where global minima correspond exactly to valid codewords. The decoder operates as a codeword projector, using gradient dynamics to navigate the continuous relaxation of the binary error space. Key Contributions The Energy Landscape: Formalization of the squared-spring Hamiltonian H(v)H(v)H(v), proving that valid codewords are the only points with zero energy and zero gradient (Theorem 3.4). Basin Geometry & Convergence: Analytical derivation of the exact Hessian and gradient at valid codewords (Theorems 4.1, 4.3). The work defines three nested radii (rlocr_{loc}rloc​, rbarrierr_{barrier}rbarrier​, rempr_{emp}remp​) that characterize the decoder’s convergence behavior. The Girth–Trap Hypothesis: An empirical finding that challenges existing intuition; data demonstrates that higher girth does not necessarily prevent decoder trapping, with girth-8 codes observed to trap at higher rates than girth-4 variants. Optimal λ Heuristic: Discovery of a universal scaling law λ∗≈(3/16)dˉ\lambda^* \approx (3/16)\bar{d}λ∗≈(3/16)dˉ for the regularization parameter, verified across eight independent CSS code families. Hardware Acceleration: A high-throughput implementation leveraging PyTorch and CUDA, achieving >86,000 syndromes per second, representing a ~28,000× speedup over serial Python implementations. Contents Core Library: `continuous_relaxation_decoder.py` — A modular Python library for CSS code construction, GF(2) linear algebra, and energy-based decoding. Verification Suite: `verification_ContinuousRelaxationDecoder.py` — A self-contained script that numerically verifies every theorem and proposition in the paper to machine precision. Computational Experiments: Momentum and Langevin dynamics comparisons. Failure mode census and pseudo-codeword barrier height calculations. Causal Diamond (CD) geometry analysis and [[208, 16, 6]] distance certification. Manuscript: LaTeX source and compiled PDF of the paper. Testbed CodesThe results are benchmarked on several high-performance QLDPC codes, including: [[193, 25, 4]] D4 Causal Diamond Hypergraph Product (HGP) code. [[112, 4, (6,6)]] and [[176, 32, (3,6)]] augmented-seed codes. Various Generalised Bicycle and CSS-LDPC code families.
Although high-threshold and low-overhead quantum low-density parity-check (qLDPC) codes, such as bivariate bicycle (BB) codes, can reduce the physical-qubit cost by an order of magnitude compared to the Kitaev toric code, their torus layout remains difficult for physical implementation. In this work, we introduce the first systematic procedure to convert BB codes into fully planar, open-boundary qLDPC codes, preserving their performance. We present planar code families with logical dimensions 6 ≤ k ≤ 13 , e.g., ⟦ 78 , 6 , 6 ⟧ , ⟦ 107 , 7 , 7 ⟧ , ⟦ 268 , 8 , 12 ⟧ , ⟦ 405 , 9 , 15 ⟧ , ⟦ 348 , 10 , 13 ⟧ , ⟦ 450 , 11 , 15 ⟧ , ⟦ 386 , 12 , 12 ⟧ , and ⟦ 362 , 13 , 11 ⟧ , all with geometrically local weight-6 stabilizers. Allowing weight-8 stabilizers produces a ⟦ 282 , 12 , 14 ⟧ code, exhibiting an efficiency metric ( k d 2 / n ) an order of magnitude higher than the surface code. The construction combines boundary-anyon condensation with the “lattice-grafting” optimization, yielding high-performance qLDPC codes natively compatible with planar hardware architectures. It also uncovers Sierpinski-type fractal logical operators the distance of which scales with the fractal area on finite lattices. These planar qLDPC codes provide an implementable route to resource-efficient, high-threshold fault tolerance and a flexible framework for future code design on realistic two-dimensional hardware.
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.