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

TL;DR

This paper develops a numerical method combining genetic algorithms and topological quantum theory to construct universal quantum gate sets for SU(2)k anyon models, demonstrating their computational universality.

Contribution

It introduces a genetic algorithm-enhanced approach to synthesize universal quantum gates for SU(2)k anyon models, verifying their potential for topological quantum computation.

Findings

Successfully constructed universal gate sets for SU(2)5, SU(2)6, and SU(2)7 models.

Verified the universal quantum computation capability of these anyon models.

Achieved exact implementation of the local equivalence class [SWAP] with nine EBMs.

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.