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

Quantum error correction is essential for scaling quantum computers, but finding high-performing quantum low-density parity-check (qLDPC) codes—which offer finite encoding rates and growing distance—is a difficult combinatorial design problem. The authors seek to automate the discovery of these codes by using large language models (LLMs) to navigate the vast, discrete space of algebraic construction schemes.

The authors propose Structured Concept Evolution (SCE), a framework that treats code discovery as an evolutionary search over high-level algebraic concepts. Instead of asking an LLM to design a specific parity-check matrix, the LLM acts as a mutation operator that modifies the underlying algebraic specifications (such as group-algebra elements, protograph geometry, and base space) of lifted-product codes. These concepts are paired with executable programs that generate the actual parity-check matrices for any block length. The search maintains a quality-diversity archive, using score-conditioned prompting to bias the LLM toward better-performing constructions while maintaining diversity in the code families.

SCE successfully discovered a wide range of competitive qLDPC codes, including families based on non-abelian groups that extend beyond traditional bivariate-bicycle designs. At a block length of approximately 1500, the discovered codes achieved encoding rates (k/n) up to 0.13 while maintaining logical error rates comparable to or better than established baselines. The framework proved effective even when using lightweight models (GPT-5.4-mini and GPT-5.4-nano), demonstrating that structured algebraic grammars can significantly simplify the search space for LLMs.