We compute the error threshold for the semion code, the companion of the Kitaev toric code with the same gauge symmetry group ${\mathbb{Z}}_2$. The application of statistical mechanical mapping methods is highly discouraged for the semion code, since the code is non-Pauli and non-Calderbank-Shor-Steane (CSS). Thus, we use machine learning methods, taking advantage of the near-optimal performance of some neural network decoders: multilayer perceptrons and convolutional neural networks (CNNs). We find the values $p_{\text{eff}}=9.5\%$ for uncorrelated bit-flip and phase-flip noise, and $p_{\text{eff}}=10.5\%$ for depolarizing noise. We contrast these values with a similar analysis of the Kitaev toric code on a hexagonal lattice with the same methods. For convolutional neural networks, we use the ResNet architecture, which allows us to implement very deep networks and results in better performance and scalability than the multilayer perceptron approach. We analyze and compare in detail both approaches and provide a clear argument favoring the CNN as the best suited numerical method for the semion code.

中文翻译：

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使用神经解码器确定Semion码阈值

我们计算出Semion码的误差阈值，Semion码是Kitaev Toric码的同伴，具有相同的量规对称性组 ${\mathbb{ž}}_2$。强烈建议不要将统计机械映射方法的应用用于Semion代码，因为该代码是非Pauli和非Calderbank-Shor-Steane（CSS）。因此，我们利用机器学习方法，充分利用了某些神经网络解码器的最佳性能：多层感知器和卷积神经网络（CNN）。我们找到价值$p_{\text{效}}=9.5％$ 用于不相关的位翻转和相位翻转噪声，以及 $p_{\text{效}}=10.5％$用于消除噪音。我们将这些值与使用相同方法对六角形格子上的Kitaev复曲面代码进行类似分析进行对比。对于卷积神经网络，我们使用ResNet架构，该架构允许我们实现非常深的网络，并且比多层感知器方法具有更好的性能和可伸缩性。我们对两种方法进行了详细的分析和比较，并提供了明确的论点，支持将CNN作为最适合Semion代码的数值方法。