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

Abstract:We systematically study a numerical method for constructing a universal quantum gate set for topological quantum computation (TQC) using $\mathrm{SU}(2)_k$ anyon models. The $F$-matrices and $R$-symbols are computed through the $q$-deformed representation theory of $\mathrm{SU}(2)$, enabling precise determination of elementary braiding matrices (EBMs) for $\mathrm{SU}(2)_k$ anyon systems. Quantum gates are derived from these EBMs. One-qubit gates are synthesized using a genetic algorithm-enhanced Solovay--Kitaev algorithm (GA-enhanced SKA), while two-qubit gates are constructed through brute-force search or GA optimization to approximate the local equivalence class $[\mathrm{CNOT}]$. Implementing this framework for the $\mathrm{SU}(2)_5$, $\mathrm{SU}(2)_6$, and $\mathrm{SU}(2)_7$ models successfully generates the canonical universal gate set $\{H\text{-gate},\ T\text{-gate},\ \mathrm{CNOT}\text{-gate}\}$. These numerical results provide conclusive verification of the universal quantum computation capabilities inherent in $\mathrm{SU}(2)_k$ anyon models. Furthermore, we obtain exact implementations of the local equivalence class $[\mathrm{SWAP}]$ using nine EBMs in each model.

arXiv:2505.01774 [quant-ph]

(or arXiv:2505.01774v4 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2505.01774 Focus to learn more arXiv-issued DOI via DataCite

Phys. Lett. A (2026)

https://doi.org/10.1016/j.physleta.2025.131142 Focus to learn more DOI(s) linking to related resources

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.