One of the main challenge 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 here analyze the performance of surface codes, one of the most important class for practical implementations, on both symmetric and asymmetric quantum channels. We derive approximate expressions, confirmed by simulations, to evaluate the performance of surface codes and of XZZX codes, and provide a metric to assess the advantage of codes with respect to uncoded systems. Our findings allow to characterize the performance by means of analytical formulas of surface codes, like, for example, the [[13, 1, 3]], the [[23, 1, 3/5]], the [[33, 1, 3/7]], and the [[41, 1, 5]] surface codes.

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，量子纠错码对量子计算和量子互联网的发展具有首要意义。本文分析了表面码（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。我们推导了近似表达式（经仿真验证）来评估表面码和XZZX码的性能，并提出了一个度量标准来评估编码相对于未编码系统的优势。我们的研究结果通过解析公式描述了表面码的性能，例如[[13,1,3]]、[[23,1,3/5]]、[[33,1,3/7]]和[[41,1,5]]表面码。