Large-Language-Model Discovery of Quantum LDPC Codes through Structured Concept Evolution

Source: arXiv:2606.24808 · Published 2026-06-23 · By Zidu Liu, Florian Marquardt

TL;DR

This paper addresses the challenging discrete combinatorial problem of discovering high-performance quantum low-density parity-check (qLDPC) codes for large-scale quantum error correction. qLDPC codes offer finite encoding rate and growing distance, key for practical fault-tolerant quantum computation, but their algebraic constructions have exponentially large, discrete search spaces and no closed-form optima. The authors introduce structured concept evolution (SCE), a novel search framework that combines a lightweight large language model (LLM) with a hierarchical structured grammar operating on algebraic specifications and executable code to evolve families of lifted-product CSS qLDPC codes. Instead of generating codes from scratch, the LLM proposes coherent mutations at the base group, protograph shape, or local group-algebra entry level, facilitating exploration of abelian and non-abelian group families. SCE builds a quality-diversity archive of candidate code families, which are evaluated under code-capacity depolarizing noise using belief propagation plus ordered statistics decoding (BP+OSD). Running SCE with GPT-5.4-mini and -nano models over 3000 iterations unearthed diverse lifted-product codes, including previously unexplored non-abelian groups beyond bivariate-bicycle codes, with block length around 1500 qubits and encoding rates up to ~0.13. Benchmarks indicate these discovered codes achieve superior logical error suppression relative to established bivariate-bicycle baselines at similar or higher rates, demonstrating SCE’s ability to efficiently search the combinatorial algebraic design space and generate practically competitive quantum LDPC codes.

Key findings

• Structured Concept Evolution (SCE) discovered lifted-product qLDPC codes at block lengths n ≈ 1500 with encoding rates up to k/n ≈ 0.13.
• Discovered codes include non-abelian group families (e.g., Dic11, D22) not previously explored in lifted-product qLDPC literature.
• Quality-diversity archive search over 3000 candidates improves best fitness score from 0.17 to 0.62 within three 1000-iteration rounds (Fig. 2b).
• Logical error rates ϵL of top SCE codes at physical error rate p = 0.06 reach ~10^-5, outperforming bivariate-bicycle (BB) codes with similar or lower rates and block lengths (Fig. 3).
• Upper bounds on code distances d computed via QDistRnd show kd^2/n values up to 51.9 for discovered codes, indicating strong finite-length protection (Table I).
• Hierarchical mutations at three levels (local group-algebra entry reseeding, protograph shape rescaling, base group substitution) enable both fine and large jumps in code parameters and fitness scores.
• Use of lightweight LLMs (GPT-5.4-mini/nano) as mutation operators enables scalable, interpretable algebraic exploration without requiring large-scale models or direct code generation.
• The evolutionary archive maintains diversity in base groups, code rates, and decoding performance, avoiding convergence to a single solution.

Threat model

Not applicable; the paper focuses on code design and error suppression in fault-tolerant quantum computing rather than security against adversaries.

Methodology — deep read

1. Threat model & assumptions: This is a discovery paper focusing on improving quantum LDPC code constructions for fault-tolerant quantum computing, rather than adversarial security. The goal is to find codes with good finite-length distance and decoding performance under code-capacity depolarizing noise. An adversary is not explicitly modeled.

2. Data: The search space consists of lifted-product CSS quantum codes constructed from two small sparse matrices A and B with entries in a finite group algebra F2[G]. The groups G explored include abelian direct products (e.g., Z2 × Z2 × Zt) and non-abelian groups like dihedral groups (Dicm). The search evolves structured algebraic specifications Σ = (G, protograph shape, local seed matrices) paired with executable programs PΣ that generate the full parity-check matrices HX, HZ at any block length scaling n = f·|G|.

3. Architecture / Algorithm: SCE is an evolutionary search framework where individuals are these structured concept specifications+programs. Mutations occur hierarchically at three levels: (i) level-1 local changes to entries of A and B keeping the group and protograph fixed (local reseeding), (ii) level-2 protograph shape and scaling mutations that fix G but change dimensions mA×nA, mB×nB altering block length and rate, and (iii) level-3 algebraic mutations that replace the base group G with a different group, adjusting related data accordingly. Each mutation proposes a valid child concept. A lightweight LLM (mixed GPT-5.4-mini and GPT-5.4-nano) is prompted with a parent concept and top-performing archive samples as in-context examples and returns a modified algebraic specification and executable.

The executable program PΣ constructs the full lifted-product parity-check matrices via regular representations of group-algebra elements and protograph rules, ensuring the CSS commutation condition by construction.

4. Training regime: The evolutionary search runs in three sequential rounds of 1000 iterations each, sampling mutation levels uniformly at random in each iteration. The archive maintains a quality-diversity population filtered by novelty (to avoid near duplicates) and fitness score (Eq. 2) combining logical qubit count and empirical error suppression exponent α(p). The fitness score is computed from logical error rates estimated through Monte-Carlo sampling using BP+OSD decoding at physical depolarizing noise error rates P = {0.05, 0.06, 0.07}.

5. Evaluation protocol: Each child candidate code is checked for CSS validity and theoretically verified. Codes that encode at least one logical qubit and have stabilizer weight ≤14 are decoded under the depolarizing noise using BP+OSD decoder with min-sum updates, 2000 (search) or 5000 (final eval) iterations, plus ordered statistics post-processing. Logical error rates are estimated with 2000 to 10^6 samples per physical error rate p. The fitness is the score S combining rate and suppression exponent, weighted by a novelty factor penalizing close duplicates in graph fingerprint space (Weisfeiler-Lehman graph kernel). At the end of each round, the 10 top performers at p=0.06 seed the next round. Final benchmarking compares elite SCE codes against standard bivariate-bicycle codes at similar block lengths.

6. Reproducibility: SCE is implemented atop OpenEvolve, an open-source evolutionary coding agent. The code and supplemental materials provide grammar details and archival parameters. Datasets are synthetic, generated programmatically by the executable programs derived from group algebra and protograph parameters, so the data is closed-form and exactly reproducible given the code and seeds. No frozen LLM weights need to be released beyond OpenAI GPT-5.4-mini/nano usage.

Example End-to-End: Starting from seed lifted-product codes parameterized by group G (e.g., Z2×Z2×Zt), matrices A and B, and protograph dimensions, the LLM proposes an algebraic mutation. This could be a level-2 mutation reshaping the protograph (change from 3×4 to 5×3) with new seed entries for A, B. The program PΣ updates to generate new (HX, HZ) at block length ~1500. The code is decoded by BP+OSD at sample error rates, and the logical error rate estimates are used to compute fitness S. Promising codes enter the archive for future sampling and mutation. Over rounds, higher fitness codes emerge with improved error suppression and rate.

Technical innovations

• Introduction of Structured Concept Evolution (SCE), a hierarchical mutation grammar operating on algebraic code specifications paired with executable construction programs, rather than raw matrices.
• Use of lightweight large language models (GPT-5.4-mini/nano) as intelligent mutation operators guiding evolutionary search in a highly structured discrete combinatorial algebraic domain.
• The three-level mutation strategy (local reseeding, protograph shape rescaling, base group substitution) enabling coherent jumps across code families while preserving CSS validity by construction.
• Quality-diversity archive using graph-fingerprint novelty metrics combined with logical error rate fitness to maintain and progressively improve a diverse set of lifted-product quantum LDPC code constructions.
• Demonstration of discovering non-abelian lifted-product qLDPC code families with finite length block size ~1500 and encoding rate ~0.13, outperforming existing bivariate-bicycle baseline codes under depolarizing noise.

Baselines vs proposed

• Bivariate-bicycle codes (BB [[144,12,12]], [[288,12,18]], [[360,12,≤24]], [[756,16,≤34]]): logical error rate (ϵL @ p=0.06) ~10^-4 to 10^-3 vs SCE discovered codes: ϵL ~10^-5 at similar or higher k/n ≈ 0.13 (Fig. 3a,b)
• SCE archive evolution: fitness score improved from 0.17 to 0.62 over 3000 candidate evaluations (Fig. 2b).
• QDistRnd upper bounds on distance: SCE codes achieve kd^2/n up to 51.9 vs prior finite-length enumerations with lower distance bounds (Table I).

Limitations

• The minimum distance values are upper bounds found by heuristic random decoding methods (QDistRnd) rather than exact distance, which is NP-hard.
• Performance evaluation is limited to code-capacity depolarizing noise; no adversarial or circuit-level noise or fault models considered.
• No investigation of robustness under hardware connectivity constraints or gate-level compilation overheads.
• Search uses relatively small maximum block lengths (up to 1500 qubits); asymptotic scaling behaviors remain unverified experimentally.
• LLM mutations rely on prompt engineering with in-context examples and manual seed programs, which may bias the search space explored.
• The decoder used (BP+OSD) is standard but not optimized, and stronger decoding methods or real-time adaptive decoders were not explored.

Open questions / follow-ons

• Can SCE be extended to incorporate hardware-specific constraints such as qubit connectivity graphs and syndrome measurement circuits?
• Could the approach be generalized to discover non-lifted-product quantum codes such as balanced product, fiber-bundle, or quantum Tanner codes?
• Is it possible to co-design encoding circuits and decoding procedures jointly with the code using SCE as an end-to-end automatic search framework?
• Can higher-capacity or more expressive LLMs improve search efficiency or discover fundamentally novel CSS-preserving constructions beyond current families?

Why it matters for bot defense

While this work is specialized for quantum error-correcting code design, the structured evolutionary approach combined with LLM-guided mutations illustrates a powerful methodology for navigating complex discrete combinatorial spaces in security-critical domains. Bot-defense and CAPTCHA systems often face similarly vast, structured design spaces for challenge constructions or detection heuristics. Adapting a hierarchical mutation grammar operated via lightweight LLMs could enable discovery of novel CAPTCHA schemes or adversarially robust detection features by evolving interpretable concepts rather than raw instances. Additionally, quality-diversity archives supporting diverse candidate generation could help defense practitioners balance security, usability, and novelty. The approach of combining symbolic algebraic representations with executable semantics may inspire more principled, scalable design tools in bot-defense beyond brute-force or purely heuristic methods. However, concrete application requires adapting the mutation grammars and fitness functions to CAPTCHA-specific challenges such as human interaction modeling or adversarial bot capabilities.

Cite

@article{arxiv2606_24808,
  title={ Large-Language-Model Discovery of Quantum LDPC Codes through Structured Concept Evolution },
  author={ Zidu Liu and Florian Marquardt },
  journal={arXiv preprint arXiv:2606.24808},
  year={ 2026 },
  url={https://arxiv.org/abs/2606.24808}
}

Read the full paper

• arXiv:2606.24808 (abstract)
• arXiv:2606.24808 (PDF)