This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.

翻译：本文聚焦于泡利信道的错误阈值问题。我们采用DiVincenzo、Shor和Smolin于1998年开创的陪集权重枚举子解析框架，通过数值计算给出了阈值的下界。具体而言，我们研究了多种小型稳定子码及其级联码的潜在非可加性，并报告了若干展现显著非可加性的短长度级联稳定子码。同时，我们给出了级联相位翻转与比特翻转重复码的陪集权重枚举子闭式表达式。基于该形式化方法获得的洞见，我们估算了长长度级联重复码的阈值。最后，针对多种小型稳定子码的级联结构，我们优化了在对应信道哈希点处产生最大非可加性的信道参数。此外，我们从非可加性与阈值问题的视角，对各类稳定子码的性能表现进行了补充讨论。本文报告了正向与负向结果，并着重指出了一些反直觉的观测现象，以期为后续错误阈值下界研究提供支撑。