Liang Zhipeng, Wang Zicheng, Yi Zhengzhong, Yang Fusheng, Wang Xuan
School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China.
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China.
Entropy (Basel). 2025 Sep 9;27(9):940. doi: 10.3390/e27090940.
The code distance is a critical parameter of quantum stabilizer codes (QSCs), and determining it-whether exactly or approximately-is known to be an NP-complete problem. However, its upper bound can be determined efficiently by some methods such as the Monte Carlo method. Leveraging the Monte Carlo method, we propose an algorithm to compute the upper bound on the code distance of a given QSC using fully decoupled belief propagation combined with ordered statistics decoding (FDBP-OSD). Our algorithm demonstrates high precision: for various QSCs with known distances, the computed upper bounds match the actual values. Additionally, we explore upper bounds for the minimum weight of logical operators in the Z-type Tanner-graph-recursive-expansion (Z-TGRE) code and the Chamon code-an XYZ product code constructed from three repetition codes. The results on Z-TGRE codes align with theoretical analysis, while the results on Chamon codes suggest that XYZ product codes may achieve a code distance of O(N2/3), which supports the conjecture of Leverrier et al.
码距是量子稳定器码（QSCs）的一个关键参数，已知确定其精确值或近似值是一个NP完全问题。然而，其上限可以通过一些方法（如蒙特卡罗方法）有效地确定。利用蒙特卡罗方法，我们提出了一种算法，该算法使用完全解耦的置信传播与有序统计解码（FDBP-OSD）相结合来计算给定QSC的码距上限。我们的算法展示了高精度：对于各种已知距离的QSCs，计算出的上限与实际值相匹配。此外，我们探索了Z型 Tanner 图递归扩展（Z-TGRE）码和Chamon码（一种由三个重复码构造的XYZ乘积码）中逻辑算子最小权重的上限。Z-TGRE码的结果与理论分析一致，而Chamon码的结果表明XYZ乘积码可能实现O(N2/3)的码距，这支持了勒维里尔等人的猜想。
