Determining the upper bound of code distance of quantum stabilizer codes through Monte Carlo method based on fully decoupled belief propagation

Code distance is an important parameter for quantum stabilizer codes (QSCs). Directly precisely computing it is an NP-complete problem. However, the upper bound of code distance can be computed by some efficient methods. In this paper, employing the idea of Monte Carlo method, we propose the algorithm of determining the upper bound of code distance of QSCs based on fully decoupled belief propagation. Our algorithm shows high precision - the upper bound of code distance determined by the algorithm of a variety of QSCs whose code distance is known is consistent with actual code distance. Besides, we explore the upper bound of logical X operators of Z-type Tanner-graph-recursive-expansion (Z-TGRE) code and Chamon code, which is a kind of XYZ product code constructed by three repetition codes. The former is consistent with the theoretical analysis, and the latter implies the code distance of XYZ product codes can very likely achieve $O(N^{2/3})$, which supports the conjecture of Leverrier et al..