Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation

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 <i>X</i> 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.