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

TL;DR

The paper introduces a human-in-the-loop, multi-agent workflow for co-designing quantum error-correcting codes with prescribed diagonal transversal gates, built on the SSLP framework and implemented in TeXRA. By separating residue-based combinatorial screening from linear Z-marginal feasibility, the approach converts a nonlinear constraint problem into tractable subproblems, enabling large-scale enumeration and exact reconstruction of codes with distance $d=2$ for $n\,\le\,6$ and $K\in\{2,3,4\}$. The Synthesis, Search, and Audit agents collaboratively derive closed-form analytical families from verified instances, and demonstrate extensions beyond nondegenerate residues to a $((6,4,2))$ code implementing diag$(1,1,1,i)$. The workflow shows how AI-assisted discovery, coupled with rigorous independent audit, can generate reproducible code constructions, push toward higher distances and larger $K$, and foster data-driven classification of transversal gate capabilities in small quantum codes.

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 $K\in\{2,3,4\}$ and $n\le6$ 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.

Figures (2)

• Figure 1: Schematic diagram of the human-AI co-design workflow for quantum code discovery. The researcher seeds the system with basic definitions and a worked example (Sec. \ref{['sec:formulation']}), reviews outputs, and provides checkpoints (orange). The Synthesis Agent deduces a combinatorial reformulation of the problem and proposes parameter templates (Sec. \ref{['subsec:synthesis']}). Guided by these proposals, the Search Agent generates and executes search algorithms on a local workstation to enumerate candidate codes (black; Sec. \ref{['subsec:search_agent']}). The resulting candidates are passed to the Audit Agent, which, along with the researcher, independently verifies that they satisfy the exact KL and transversal conditions (Sec. \ref{['subsec:audit']}). Verified results are fed back to the Synthesis Agent to abstract general code families and construct extended examples (Secs. \ref{['sec:results']}, \ref{['sec:families']}). Legend: Blue arrows denote automated agent-to-agent flow; green arrows indicate human-AI communication; orange arrows mark human oversight; the red cross, $\textcolor{red}{X}$, highlights the intentional "no-communication" barrier that preserves the independence of the audit process.
• Figure 2: Two types agent supported by TeXRA operating on a shared workspace (LaTeX files, Python scripts, data) synchronized to a remote repository (GitHub/Overleaf). Both communicate with GPT-5 via API. (a) Derivation-then-edit workflow: Round 0 derives, Round 1 extends reasoning (test-time scaling OpenAI2024OpenAI) and generates LaTeX edits; human instructs and reviews output. (b) Tool-use loop: Iterative ReAct cycle (reason $\to$ act $\to$ observe) where act means writing/executing code; human discusses throughout. Bidirectional arrows show interactive communication.