One of the main challenges for an efficient implementation of quantum information technologies is how to counteract quantum noise. Quantum error correcting codes are therefore of primary interest for the evolution towards quantum computing and quantum Internet. We analyze the performance of stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors of stabilizer codes based on the quantum MacWilliams identities. The WE is then used to evaluate the error rate of quantum codes under maximum likelihood decoding or, in the case of surface codes, under minimum weight perfect matching (MWPM) decoding. Our findings lead to analytical formulas for the performance of generic stabilizer codes, including the Shor code, the Steane code, as well as surface codes. For example, on a depolarizing channel with physical error rate $\rho \to 0$ it is found that the logical error rate $\rho_\mathrm{L}$ is asymptotically $\rho_\mathrm{L} \to 16.2 \rho^2$ for the $[[9,1,3]]$ Shor code, $\rho_\mathrm{L} \to 16.38 \rho^2$ for the $[[7,1,3]]$ Steane code, $\rho_\mathrm{L} \to 18.74 \rho^2$ for the $[[13,1,3]]$ surface code, and $\rho_\mathrm{L} \to 149.24 \rho^3$ for the $[[41,1,5]]$ surface code.

翻译：量子信息技术高效实现的主要挑战之一是如何对抗量子噪声。因此，量子纠错码对于量子计算和量子互联网的发展具有首要意义。我们分析了稳定子码（实际应用中最重要的一类码）在对称和非对称量子信道上的性能。为此，我们首先基于量子MacWilliams恒等式推导了稳定子码不可检测错误的重量枚举子（WE）。随后，利用该WE评估了量子码在最大似然译码下，或对于表面码而言在最小权重完美匹配（MWPM）译码下的错误率。我们的研究得出了通用稳定子码（包括Shor码、Steane码以及表面码）性能的解析公式。例如，在物理错误率为$\rho \to 0$的退极化信道上，发现逻辑错误率$\rho_\mathrm{L}$渐近趋于：对于$[[9,1,3]]$ Shor码为$\rho_\mathrm{L} \to 16.2 \rho^2$，对于$[[7,1,3]]$ Steane码为$\rho_\mathrm{L} \to 16.38 \rho^2$，对于$[[13,1,3]]$表面码为$\rho_\mathrm{L} \to 18.74 \rho^2$，而对于$[[41,1,5]]$表面码则为$\rho_\mathrm{L} \to 149.24 \rho^3$。