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).
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).
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 (E8 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∼100–200 physical qubit regime. Scientific Contributions The Augmented-Seed Code Family: The central discovery that a specific weight-3 row (one of 64 in two B4 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≥67 for the `[[193, 25]]` code are formally retracted; an exact Integer Linear Program (ILP) oracle is introduced, definitively proving d=4. The E8 Structural Obstruction (Theorem 4.5): Rigorous proof that the 28-qubit E8 Lorentzian code cannot be punctured to yield a valid quantum code. Its 7-disconnected-4-cycle geometry guarantees dZ=1 after any single row drop, pointing toward the Euclidean E8 root lattice as the necessary next step. Naïve CSS Lift Theorem (Theorem 5.3): Algebraic proof that setting HX=ker(HZ) 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 kd2/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 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.
Error correction thresholds are often treated as the primary figure of merit for comparing quantum error-correcting code families. We show that the optimal error correction threshold for many commonly considered codes is constrained to a single universal value at leading order in a replica limit. Through a statistical mechanical mapping, we demonstrate that duality constrains all zero-rate em-symmetric CSS codes to have the same optimal code capacity threshold. Here, em symmetry means that the X- and Z-type parity-check matrices are equivalent up to row and column permutations. Under this statistical mechanical mapping, em-symmetric CSS codes are self-dual under a generalized Kramers-Wannier duality up to a mixing of logical sectors. For zero-rate code families, this mixing contributes only subextensive corrections, so the thermodynamic bulk free energy is self-dual in the trivial logical sector. This self-duality fixes the clean critical point and constrains the disordered phase boundary. We also show that self-duality is preserved under code concatenation, and that optimal decoding of concatenated codes can be reformulated as a renormalization group flow on a hierarchical lattice. Our results provide a common framework for analyzing topological, concatenated, and more general quantum low-density parity-check code families, including both their optimal code capacity thresholds and their sub-threshold logical error suppression.
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 (rloc, rbarrier, 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 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.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group G and two subsets of G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%, approximately matching that of the [[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
Quantum error correction (QEC) codes protect quantum information from errors due to decoherence. Many of them also serve as prototypical models for exotic topological quantum matters. Investigating the behavior of the QEC codes under decoherence sheds light on not only the codes' robustness against errors but also new out-of-equilibrium quantum phases driven by decoherence. The phase transitions, including the error threshold, of the decohered QEC codes can be probed by the systems' R\'enyi entropies SR with different R\'enyi indices R. In this paper, we study the general construction of the statistical models that characterize the R\'enyi entropies of QEC codes decohered by Pauli noise. We show that these statistical models can be organized into a ``tapestry'' woven by rich duality relations among them. For Calderbank-Shor-Steane (CSS) codes with bit-flip and phase-flip errors, we show that each R\'enyi entropy is captured by a pair of dual statistical models with randomness. For $R=2,3,\ensuremath{\infty}$, there are additional dualities that map between the two error types, relating the critical bit-flip and phase-flip error rates of the decoherence-induced phase transitions in the CSS codes. For CSS codes with an ``em symmetry'' between the X-type and the Z-type stabilizers, the dualities with $R=2,3,\ensuremath{\infty}$ become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by ${S}_{R=2,3,\ensuremath{\infty}}$. For general stabilizer codes decohered by generic Pauli noise, we also construct the statistical models that characterize the systems' entropies and obtain general duality relations between Pauli noise with different error rates.
This paper establishes the algebraic and geometric structure of the [[13,7,3]]_3 ternary CSS quantum stabilizer code and its dimension-indexed family, arising from the Topological Vortex Logic (TVL) framework introduced in "T³ as a Closed Information-Processing Environment" (Zenodo DOI: 10.5281/zenodo.19682633). The 26 stable winding states of TVL on the three-torus T³ form 13 antipodal pairs {w, −w}. These pairs are in natural bijection with the 13 points of the projective plane P²(F₃), and the Z₃ charge structure of TVL induces a parity-check matrix making the antipodal pairs the physical qutrits of the [[13,7,3]]_3 ternary CSS quantum Hamming code (Ketkar et al. 2006). This paper studies the structural properties of this code and its dimension-indexed family [[(3^d−1)/2, (3^d−1)/2 − 2d, 3]]_3 for d ≥ 2. Ten structural results are established: (1) Family construction and self-orthogonality: The CSS construction is valid at every level d ≥ 2, with H·H^T = 0 over F₃ proved for all d. (2) Double distinction of d=3: The TVL code is the smallest family member with positive logical dimension and the unique member among d ≤ 6 with prime physical qutrit count (n=13; the next occurs at d=7, n=1093). (3) PSL irreducibility (iff d=3): PSL(d,F₃) acts absolutely irreducibly on the logical space L_d = C_d/C_d^⊥ if and only if d=3 — proved at d=3 via Burnside's criterion (the 7-dimensional logical module). For every d ≥ 4 the action is reducible: L_d is a uniserial F₃[SL(d,F₃)]-module with exactly d−2 composition factors, by the Bardoe–Sin submodule structure of the GL(n,F_q) permutation module on PG(n−1,q). The earlier converse-of-Schur argument fails in defining characteristic, where dim End_G = 1 does not imply irreducibility. The logical space exists only on the self-orthogonal range q ∈ {2,3}, where the same d−2-factor structure holds in both fields — the binary factors being the trivial module and the exterior powers Λ^k V of the natural module — so that L_d is irreducible exactly for (q,d) ∈ {(3,3),(2,3)}: the TVL code and the Steane code [[7,1,3]] (Proposition 7). (4) σ-structural properties of the family: The shift automorphism σ_d is an isometry of F₃^(n_d) for the weight form; its eigenspaces are mutually symplectically orthogonal; its maximum Jordan block size on the logical space L_d equals 3^(ν₃(d)) (the 3-adic valuation of d); the σ-fixed Gram matrix has rank 1; and the construction is self-orthogonal over F_q if and only if q ∈ {2,3} (the Q-Scope theorem, via a Gauss power-sum identity). (5) Stabilizer weight enumerator: The distribution {A₀=1, A₉=104, A₁₂=624} is derived entirely from the line-incidence geometry of P²(F₃), with each sub-case verified independently. (6) Closed-form weight enumerator: A(ω) = i^(d mod 2) · 3^(n_d/2) for all d ≥ 2, with |A(ω)|² = 3^(n_d) an exact integer. The formula is specific to q=3: the magnitude equality |1+(q−1)ω| = |1−ω| = √3 holds uniquely at q=3. (7) Quantum weight enumerators: The Shor–Laflamme stabilizer and normalizer weight enumerators of the family are given in closed form. The stabilizer enumerator A_d(z) generalizes the d=3 line-incidence count to hyperplanes in PG(d−1, F₃); the normalizer enumerator B_d(z) is its F₉ MacWilliams transform; and the minimum distance d_min = 3 is recovered directly. (8) Non-standard quadratic form: The weight-mod-3 function on C_d/C_d^⊥ defines a non-degenerate quadratic form of Witt class +1 for d ∈ {3,4,5}. At d=3 this differs from the standard sum-of-squares form on F₃^7 (Witt class −1); at d=4 both forms coincide at Witt class +1. (9) Shell stratification as level sets: TVL's face, edge, and corner shells (|w|²=1, 2, 3) correspond precisely to the level sets Q₃=1, 2, 0 of the standard quadratic form on the parity-check space F₃³. (10) Polar duality and parabolic nesting: The q₃=0 sector of TVL is the polar of the corner point (1,1,1) with respect to the absolute conic. The q₃=0 line is σ-invariant. The family nesting d→d+1 corresponds to maximal parabolic subgroup actions in PSL(d+1,F₃). The paper applies the automorphism-group-action framework of Grassl and Rötteler (2013) to the ternary projective Hamming family and identifies three structural features not studied in that context: the closed-form weight enumerator, the family-wide Witt-class result, and the q=3 exceptionality identity. Five Python verification scripts with accompanying README are included as ancillary files, reproducing the computational results cited in the paper. Published in conjunction with the TVL framework archived at: 10.5281/zenodo.19682633 (Concept DOI, always resolves to latest version). Note on this record. Every computational claim in this version is reproduced from first principles by the five ancillary scripts, and the central results have each been re-derived by an independent method. The revisions reflect that standard: open characterization questions were resolved one by one, and the result shown to be overstated — the PSL-irreducibility characterization — was corrected in the open, sharpening the paper's central claim that [[13,7,3]]₃ is the unique ternary code in its family with an irreducible logical space. Version 1.0.7 is the current, self-contained version. Changelog v1.0.7 (June 2026) Self-contained version: the version-history scaffolding is removed and one result is added to Section 4.1. Version-history scaffolding removed from the paper. The version note (which documented the v1.0.5 PSL-irreducibility correction and the v1.0.6 weight-enumerator additions) is removed, and the title page no longer carries a version marker or version-note pointer; the paper now reads as a standalone work. The full revision history is retained in this changelog (below) and in the Zenodo version record.2. Section 4 — composition structure across the self-orthogonal range (Proposition 7, new). The former "General q" remark — which had incorrectly stated that L_d = C_d/C_d^⊥ is defined for general q and that the ternary d=3 code is the unique irreducible instance — is replaced by Proposition 7: L_d exists only for q ∈ {2,3} (the self-orthogonal range); in both fields it has exactly d−2 composition factors (for q=3 of dimensions [x^j](1+x+x²)^d; for q=2 the trivial module together with the exterior powers Λ^k V of the natural module); and it is irreducible exactly for (q,d) ∈ {(3,3),(2,3)} — the TVL code [[13,7,3]]₃ and the Steane code [[7,1,3]]. The q=2 structure is classical (Bardoe–Sin for all q; the exterior filtration of the elementary-abelian group algebra) and is cited, not claimed as new. The dim End_G(L_d) = 1 remark — explaining why this does not force irreducibility in defining characteristic (3 | |SL(d,F₃)|, so L_d need not be semisimple; at d=4 reducible-but-indecomposable with End_G(L_d) = F₃) — is retained as a standalone caution. A previously unnumbered remark becomes a numbered proposition (Proposition 7), so the subsequent theorems, propositions, and corollaries shift up by one — for example the closed-form weight-enumerator theorem is now Theorem 14 and the family-wide Witt-class theorem is now Theorem 19 (it was Theorem 18 in v1.0.6). All cross-references are symbolic and update automatically; the ancillary verification scripts and README are updated to the new numbers. The paper remains 19 pages and compiles with no undefined references.3. The open-questions section is streamlined to the single genuinely open item, physical realisation; the questions resolved in earlier revisions (the closed-form classical and quantum weight enumerators, the family-wide d≥4 minimum distance, general PSL irreducibility, and now the general-q module structure via Proposition 7) are presented as results in the body.4. Conclusions item 3 is extended to record the two-field characterization and the Steane instance.5. Verification scripts: a fifth ancillary script, verify_logical_both_fields.py, is added; it reproduces the Proposition 7 q=2 composition factors, the well-posedness q ∈ {2,3}, and the irreducibility criterion by an independent method (exterior filtration and short exact sequences). The four earlier scripts are unchanged.6. Title page and date updated to v1.0.7. v1.0.6 (June 2026) Completes the weight-enumerator treatment and refines the open questions. This revision adds results: subsequent theorems and propositions are renumbered upward (for example the family-wide Witt-class theorem is now Theorem 18), with all cross-references symbolic and updating automatically.1. Classical weight enumerator (Theorem 13). The closed-form weight enumerator of the family is given as the complete polynomial W_d(z), with its value at omega = e^{2 pi i/3} — the Gauss-sum value i^{d mod 2} 3^{n_d/2} — recorded as Corollary 14. The d=3 expansion is given explicitly; its coefficient of z^11 is |PSL(3,3)| = 5616.2. Quantum weight enumerators (new Section 6.4). The Shor–Laflamme stabilizer and normalizer weight enumerators of the family are given in closed form (Propositions 15 and 16). The stabilizer enumerator A_d(z) generalizes the d=3 line-incidence count of the stabilizer-enumerator proposition (Section 6.1) to hyperplanes in PG(d−1,3); the normalizer enumerator B_d(z) is its F_9 MacWilliams transform; and the minimum distance d_min = 3 is recovered directly from the pair (Corollary 17).3. Open Question 2 closed. Items 1 and 2 together close the open question on the closed-form classical and quantum weight enumerators for the family.4. Open Question 1 narrowed (general q). The general q = p^t module structure is determined by Bardoe–Sin (already cited in the paper): Frobenius-twisted composition factors, irreducible only in special cases, the prime-field case uniserial; the genuinely remaining item is a self-contained description of the logical subquotient C_d/C_d^⊥ for general q.5. Open Question 5 closed (minimum distance). The family-wide d_min = 3 is established in Ketkar, Klappenecker, Kumar, and Sarvepalli (2006), and is now also exhibit