Surface codes are versatile quantum error-correcting codes known for their planar geometry, making them ideal for practical implementations. While the original proposal used Pauli $X$ or Pauli $Z$ operators in a square structure, these codes can be improved by rotating the lattice or incorporating a mix of generators in the XZZX variant. However, a comprehensive theoretical analysis of the logical error rate for these variants has been lacking. To address this gap, we present theoretical formulas based on recent advancements in understanding the weight distribution of stabilizer codes. For example, over an asymmetric channel with asymmetry $A=10$ and a physical error rate $p \to 0$, we observe that the logical error rate asymptotically approaches $p_\mathrm{L} \to 10 p^2$ for the rotated $[[9,1,3]]$ XZZX code and $p_\mathrm{L} \to 18.3 p^2$ for the $[[13,1,3]]$ surface code. Additionally, we observe a particular behavior regarding rectangular lattices in the presence of asymmetric channels. Our findings demonstrate that implementing both rotation and XZZX modifications simultaneously can lead to suboptimal performance. Thus, in scenarios involving a rectangular lattice, it is advisable to avoid using both modifications simultaneously. This research enhances our theoretical understanding of the logical error rates for XZZX and rotated surface codes, providing valuable insights into their performance under different conditions.

翻译：表面码是一类以平面几何结构著称的通用量子纠错码，非常适合实际应用。原始方案采用方形结构中的泡利$X$或泡利$Z$算子，但通过旋转晶格或在XZZX变体中混合生成元可改进这些码。然而，这些变体的逻辑错误率缺乏全面的理论分析。为填补这一空白，我们基于稳定子码重量分布的最新进展，提出了理论公式。例如，在不对称因子$A=10$且物理错误率$p \to 0$的非对称信道上，旋转$[[9,1,3]]$ XZZX码的逻辑错误率渐近逼近$p_\mathrm{L} \to 10 p^2$，而$[[13,1,3]]$表面码则为$p_\mathrm{L} \to 18.3 p^2$。此外，我们观察到非对称信道中矩形晶格的特定行为。研究结果表明，同时应用旋转和XZZX改进可能导致性能次优。因此，在矩形晶格场景中，建议避免同时使用这两种改进。本研究深化了对XZZX和旋转表面码逻辑错误率的理论认识，为理解它们在不同条件下的性能提供了重要见解。