The construction of a universal quantum gate set for the SU(2)k (k=5,6,7) anyon models via genetic optimized algorithm

Abstract

We study systematically numerical method into constructing a universal quantum gate set for topological quantum computation (TQC) using SU(2)k anyon models. The F-matrices and R-symbol were computed through the q-deformed representation theory of SU(2), enabling precise determination of elementary braiding matrices (EBMs) for SU(2)k anyon systems. Quantum gates were derived from these EBMs. One-qubit gates were synthesized using a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), while two-qubit gates were constructed through brute-force search or GA optimization to approximate local equivalence classes [CNOT]. Implementing this framework for SU(2)5, SU(2)6, and SU(2)7 models successfully generated the canonical universal gate set {H-gate, T-gate, CNOT-gate}. These numerical results provide conclusive verification of the universal quantum computation capabilities inherent in SU(2)k anyon models. Furthermore, we get exact implementations of the local equivalence class [SWAP] using nine EBMs in each model.