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 codes with prescribed transversal diagonal gates. It builds on the Subset-Sum Linear Programming (SSLP) framework (arXiv:2504.20847), 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 (https://texra.ai)--a multi-agent research assistant platform that supports an iterative tool-use loop agent and a derivation-then-edit workflow reasoning agent. We work in a LaTeX--Python environment where agents reason, edit documents, execute code, and synchronize their work to Git/Overleaf. Within this workspace, three roles collaborate: a Synthesis Agent formulates the problem; a Search Agent sweeps/screens candidates and exactifies numerics into rationals; and an Audit Agent independently checks all KL equalities and the induced logical action. As a first step we focus on distance d=2 with nondegenerate residues. For code dimension K in 2,3 or 4 and for up to 6 qubits, systematic sweeps yield certificate-backed tables cataloging attainable cyclic logical groups--all realized by new codes--e.g., for K=3 we obtain order 16 at n=6. From verified instances, the Synthesis Agent abstracts recurring structures into closed-form families and proves they satisfy the KL equalities for all parameters. It further demonstrates that SSLP accommodates residue degeneracy by exhibiting a new ((6,4,2)) code implementing the transversal controlled-phase diag(1,1,1,i). Overall, the workflow recasts diagonal-transversal feasibility as an analytical pipeline executed at scale, combining systematic enumeration with exact analytical reconstruction. It yields reproducible code constructions, supports targeted extensions to larger K and higher distances, and leads toward data-driven classification.