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

2026-06-23

AI Summary

Designing quantum LDPC codes that simultaneously achieve sparse parity checks, finite code rates, and growing distance remains a significant challenge. This work proposes a Structured Concept Evolution (SCE) framework that, for the first time, integrates lightweight large language models with algebraic mutation grammars to automatically discover novel lifted-product CSS quantum LDPC code families. By hierarchically evolving structured concepts in group algebras, protograph geometries, and base spaces, SCE uncovers constructions involving previously unexplored elements such as non-Abelian groups. Experimental results demonstrate that the discovered code families, when decoded under BP+OSD at code capacity, match or surpass the performance of established designs like bivariate bicycle codes, thereby validating the effectiveness and novelty of SCE in automating quantum error-correcting code design.

Abstract

Quantum computers could outperform classical machines on important problems, but only if the errors that pervade quantum hardware can be corrected at scale. Quantum low-density parity-check (qLDPC) codes offer a promising route to this goal by combining sparse parity checks with finite encoding rate and growing distance, but their construction remains a challenging discrete design problem. Here we introduce structured concept evolution (SCE), a search framework that pairs a large language model with a structured algebraic mutation grammar to discover lifted-product code families, a class of CSS qLDPC codes. Instead of asking the LLM to design codes from first principles, SCE evolves structured concepts consisting of algebraic specifications paired with executable programs that realize them, using hierarchical mutations that modify the group algebra, protograph geometry, or base space. Running SCE, we discover a diverse set of competitive code families, ranging from abelian constructions to families over non-abelian groups beyond those underlying standard designs such as bivariate-bicycle codes, and characterize them under code-capacity depolarizing noise with BP+OSD decoding. These results are obtained with lightweight models (GPT-5.4-mini and GPT-5.4-nano).

Problem

• quantum LDPC codes
• quantum error correction
• code construction
• discrete design problem
• CSS codes

Innovation

• structured concept evolution
• quantum LDPC codes
• large language models
• lifted-product codes
• algebraic mutation grammar