LLM-guided Program Evolution: A New Paradigm for Automatic Discovery of Quantum LDPC Codes

This article introduces an evolutionary workflow driven by large language models (LLMs) for the automatic discovery of bivariate bicycle codes and their perturbed variants. The method generates candidate codes by mutating Python programs via LLMs. After screening 200,000 candidate codes over approximately 140 hours of computation and with an LLM inference cost of $400, it finally discovered 465 distinct quantum codes (including 97 CSS codes and 368 non-CSS perturbed variants). This paradigm opens up a new path for quantum LDPC code discovery, combining AI with domain knowledge and applicable to complex design space exploration.

Quantum LDPC codes are the core of fault-tolerant quantum computing. Their design space is vast, making traditional manual/exhaustive methods inefficient. Bivariate bicycle codes are based on polynomial rings and have excellent structures, but their design space is still large; perturbed variants are non-CSS types with richer structures. Design faces dual challenges: parameter verification (code distance, code rate) and equivalence class identification (equivalence under local Clifford transformations).

The core innovation is using LLMs as program mutation engines, representing candidate code generators with the "program as gene" concept. The search consists of 5 activities with a total of 1650 iterations: LLMs mutate excellent programs to generate new code generators, which are executed to obtain candidate codes. The multi-stage verification pipeline: GF(2) rank calculation for fast filtering → distance estimation algorithm for preliminary evaluation → MILP for precise distance certification → BLISS for deduplication → decomposability analysis → local Clifford equivalence check, balancing efficiency and accuracy.

For block lengths n ≤ 360, 465 valid codes were discovered: 97 CSS bivariate bicycle codes and 368 non-CSS perturbed variants. Among the CSS codes, known high-performance codes were recovered, and new finite-length representatives were found (e.g., the indecomposable [[288,16,12]] code, and codes with code rate k=50 when d=8); non-CSS perturbed codes at [[144,12,12]] have performance comparable to the Gross code and offer greater design flexibility.

The total computation time was approximately 140 hours, and the LLM inference cost was about $400, resulting in a considerable input-output ratio. The LLM inference cost accounted for a small proportion; most resources were consumed in the verification phase, so improving the efficiency of the verification pipeline is a future optimization direction.

Technical significance: Demonstrates the general paradigm of "LLMs as program evolution engines", applicable to fields with complex design spaces; provides new high-performance code resources for the quantum community (especially non-CSS codes); proves that LLMs can serve as creative engines for scientific discovery. Future outlook: Optimize the efficiency of the verification pipeline, enhance LLM capabilities, and expect more breakthroughs in quantum error-correcting codes and structured design fields.

LLM-guided program evolution opens up a new path for quantum LDPC code discovery. By combining LLM code generation with rigorous mathematical verification, a large number of new quantum codes were discovered at a reasonable cost. The combination of AI and domain expertise is an effective strategy for solving complex scientific problems and will continue to drive breakthroughs in related fields in the future.