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

TL;DR

This paper introduces a multi-agent platform combining symbolic synthesis, linear programming, and formal verification in Lean to discover and verify quantum error-correcting codes with specific transversal gates.

Contribution

It extends TeXRA with Lean 4 verification, enabling exact, human-guided discovery and formal proof of quantum codes within a comprehensive AI-assisted framework.

Findings

Catalogued 14,116 quantum codes with specific properties.

Constructed a residue-degenerate ((6,4,2)) code implementing a controlled-phase gate.

Resolved the transversal-T problem for certain quantum codes, with some candidates proven impossible.

Abstract

Exact scientific discovery requires more than heuristic search: candidate constructions must be turned into exact objects and checked independently. We address this gap by extending TeXRA with an independent Lean 4 verification layer, turning it into a human-guided multi-agent platform for exact scientific discovery. The platform couples symbolic synthesis, combinatorial and linear-programming search, exact reconstruction of numerical candidates, and formal verification in Lean. We apply this platform to nonadditive quantum error-correcting codes with prescribed transversal diagonal gates within the subset-sum linear-programming (SSLP) framework. In the distance-2 regime where logical states occupy distinct residue classes, the platform yields a Lean-certified catalogue of 14,116 codes for $K\in\{2,3,4\}$ and up to six physical qubits, realizing cyclic logical orders 2 through 18, from which we extract closed-form infinite families. We also construct a residue-degenerate $((6,4,2))$ code implementing the logical controlled-phase gate $\mathrm{diag}(1,1,1,i)$. At distance 3, we resolve the transversal-$T$ problem for $((7,2,3))$ codes within the complementary binary-dihedral $\mathrm{BD}_{16}$ setting: among the 12 candidates surviving the SSLP filters, 10 admit exact realizations and 2 are excluded by no-go proofs. All accepted constructions, families, and no-go results are formalized and checked in Lean, illustrating how AI-assisted workflows can bridge search, exact reconstruction, and formal proof in the physical sciences.