Quantum error correcting codes are 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 based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. Finally, we introduce a novel approach that combines the knowledge of WE with a logical operator analysis. This method allows the derivation of the exact asymptotic performance for short 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 $\rho_\mathrm{L} \approx 16 \rho^2$ for the $[[9,1,3]]$ Shor code, $\rho_\mathrm{L} \approx 16.3 \rho^2$ for the $[[7,1,3]]$ Steane code, $\rho_\mathrm{L} \approx 18.7 \rho^2$ for the $[[13,1,3]]$ surface code, and $\rho_\mathrm{L} \approx 149.3 \rho^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $\rho_\mathrm{L} \approx 1215 \rho^4$ and $\rho_\mathrm{L} \approx 663 \rho^5$ for the $[[85,1,7]]$ and the $[[181,1,10]]$ surface codes, respectively.

翻译：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。本文分析了稳定子码（实际实现中最重要的一类码）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恒等式推导了不可检测误差的重量枚举器（WE）。随后利用该重量枚举器评估了采用最小重量译码的CSS量子码错误率的紧致上界。针对曲面码，我们还在退极化信道上导出了这些上界的简单闭合表达式。最后，我们提出了一种结合重量枚举器知识与逻辑算子分析的新型方法。该方法可推导短码的精确渐近性能。例如，在物理错误率为$\rho \to 0$的退极化信道上，发现[[9,1,3]] Shor码的逻辑错误率$\rho_\mathrm{L}$渐近为$\rho_\mathrm{L} \approx 16 \rho^2$，[[7,1,3]] Steane码为$\rho_\mathrm{L} \approx 16.3 \rho^2$，[[13,1,3]]曲面码为$\rho_\mathrm{L} \approx 18.7 \rho^2$，[[41,1,5]]曲面码为$\rho_\mathrm{L} \approx 149.3 \rho^3$。对于更大规模的码，我们的界提供了[[85,1,7]]和[[181,1,10]]曲面码的$\rho_\mathrm{L} \approx 1215 \rho^4$和$\rho_\mathrm{L} \approx 663 \rho^5$结果。