Topological quantum computing holding global anti-interference ability is realized by braiding some anyons, such as well-known Fibonacci anyons. Here, based on $\text{SO}{\left(3\right)}_2$ theory we obtain a total of six anyon models utilizing $F$ matrices, $R$ symbols, and fusion rules of metaplectic anyon. We obtain the elementary braiding matrices (EBMs) by means of unconventional encoding. After braiding $X$ and $X^{\prime }$, we insert a pair of $Z$ anyons into them to ensure that the initial order of anyons remains unchanged. In this process only fusion is required, and measurement is not necessary. Three of them $\{V_3^{113},V_3^{131},V_1^{133}\}$ are studied in detail. We study systematically the compilation of these three models through EBMs obtained analytically. For the one-qubit case, the classical $H$ and $T$ gate can be well constructed using the Genetic Algorithm-enhanced Solovay-Kitaev Algorithm (GA-enhanced SKA) by $\{V_3^{113},V_3^{131},V_1^{133}\}$. The obtained accuracy of the $H$ and $T$ gates by $\{V_3^{113},V_1^{133}\}$ is slightly inferior to the corresponding gates of the Fibonacci anyon model, but it also can meet the requirements of fault-tolerant quantum computing,with $V_3^{131}$ giving the best performance of these four models. For the two-qubit case, we use the exhaustive method for short lengths and the GA for long lengths to obtain a braidword for $\{V_3^{113},V_3^{131},V_1^{133}\}$ models. The resulting matrices can well approximate the local equivalence class of the controlled-not (cnot) gate, while demonstrating a much smaller error than the Fibonacci model, especially for the $V_3^{113}$. The braiding processes of conventional encoding (using identical anyons) and unconventional encoding (using distinct anyons) are compared. Finally, we attempt to generalize the model to the $N$-qubit case.

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• Quantum algorithms
• Quantum computation