Co-Designing Quantum Codes with Transversal Diagonal Gates via Multi-Agent Systems

We present a multi-agent, human-in-the-loop workflow that co-designs quantum error-correcting codes with prescribed transversal diagonal gates. It builds on the Subset-Sum Linear Programming (SSLP) framework, which partitions basis strings by modular residues and enforces Z-marginal Knill-Laflamme (KL) equalities via small LPs. The workflow is powered by GPT-5 and implemented within TeXRA, a multi-agent research assistant platform where agents collaborate in a shared LaTeX-Python workspace synchronized with Git/Overleaf. Three specialized agents formulate constraints, sweep and screen candidate codes, exactify numerical solutions into rationals, and independently audit all KL equalities and induced logical actions. Focusing on distance-two codes with nondegenerate residues, we catalogue new nonadditive codes for dimensions $K\in\{2,3,4\}$ on up to six qubits, including high-order diagonal transversals, yielding $14,116$ new codes. From these data, the system abstracts closed-form families and constructs a residue-degenerate $((6,4,2))$ code implementing a transversal controlled-phase $\mathrm{diag}(1,1,1,i)$, illustrating how AI orchestration can drive rigorous, scalable code discovery.