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

2025-10-23

AI Summary

This work addresses the co-design challenge of quantum error-correcting codes (QECCs) and transversal diagonal logical gates. Methodologically, it introduces the first systematic multi-agent collaborative framework, built upon an SSLP-based analytical pipeline that unifies code construction and gate realizability into scalable enumeration and analytic reconstruction problems. The framework integrates Subset-Sum linear programming, Z-edge Knill–Laflamme condition verification, and GPT-5-driven synthesis/search/audit agents within a LaTeX–Python co-execution environment (TeXRA platform), enabling synchronized reasoning, coding, and documentation. Key contributions include: (i) the first data-driven classification of cyclic logical gate groups; (ii) discovery of novel codes for $n leq 6$ qubits—including a $[![6,4,2]!]$ code realizing $mathrm{diag}(1,1,1,i)$ and a cyclic logical group of order 16 for $K=3$; and (iii) full formal verification of all results.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Adversarial Agents
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

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 $Kin{2,3,4}$ and $nle6$ 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, 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.

Problem

• Co-designing quantum codes with prescribed transversal diagonal gates
• Developing multi-agent workflow for quantum code synthesis
• Enforcing Z-marginal Knill-Laflamme equalities via linear programming

Innovation

• Multi-agent system co-designs quantum codes
• GPT-5 powers iterative tool-use loop workflow
• SSLP framework enforces KL equalities via LPs