Topological quantum compilation of metaplectic anyons based on the genetic optimized algorithms

• URL: http://arxiv.org/abs/2501.01745v4
• Date: Sun, 09 Feb 2025 09:31:19 GMT
• Title: Topological quantum compilation of metaplectic anyons based on the genetic optimized algorithms
• Authors: Jiangwei Long, Jianxin Zhong, Lijun Meng,
• Abstract summary: We obtain a total of 6 anyon models utilizing F-matrices, R-symbols, and fusion rules of metaplectic anyon. For one-qubit case, the classical H- and T-gate can be well constructed using the genetic algorithm enhanced Solovay-Kitaev algorithm.
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• Abstract: Topological quantum computing holding global anti-interference ability is realized by braiding some anyons, such as well-known Fibonacci anyons. Here, based on $SO(3)_2$ theory we obtain a total of 6 anyon models utilizing F-matrices, R-symbols, and fusion rules of metaplectic anyon.We obtain the elementary braided matrices (EBMs) by means of unconventional encoding. After braid $X$ and $X'$, we insert a pair of $Z$ anyons into they 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 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/T-gate 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, $V^3_131$ 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 braidword for $\{V_3^{113}, V_3^{131}, V_1^{133}\}$ models. The resulting matrices can well approximate the local equivalence class of the CNOT-gate, while demonstrating a much smaller error than the Fibonacci model, especially for the $V_3^{113}$.

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