Universal Quantum Gate Set from Multiple-Braiding Sequences in $SU(2)_k$ ($k>2$, $k\neq 4$) Anyon Models

Abstract

We study the implementation of a universal quantum gate set via multiple-braiding within $SU(2)_k$ ($k > 2$, $k \neq 4$) anyon models. The multiple elementary braiding matrices (MEBMs) are derived from the $q$-deformed representation theory of $SU(2)$. Braiding multiplicities from one to nine are examined as building blocks for $\{H, T, \text{CNOT}\}$ in $SU(2)_3$ and $SU(2)_5$. Only one case fails to support universality; high-precision $H$ and $T$ gates can be achieved by a Genetic Algorithm enhanced Solovay--Kitaev Algorithm, and expanding operations to 30 enables direct approximation of a locally equivalent CNOT for the remaining eight. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computing (TQC). Our numerical results provide strong evidence that most multiple-braiding sequences in $SU(2)_k$ ($k > 2$, $k \neq 4$) anyon models are capable of universal quantum computation.