Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on...

Area of Science:

• Quantum Information Science
• Quantum Error Correction
• Computational Complexity

Background:

• Determining the exact code distance of quantum stabilizer codes (QSCs) is NP-complete.
• Efficiently computing an upper bound for code distance is crucial for practical QSC applications.
• Existing methods like Monte Carlo provide a feasible approach for estimating upper bounds.

Purpose of the Study:

• To propose and validate a novel algorithm for computing the upper bound on the code distance of QSCs.
• To leverage fully decoupled belief propagation combined with ordered statistics decoding (FDBP-OSD) for this computation.
• To investigate the code distance upper bounds for specific QSCs, including Z-TGRE and Chamon codes.

Main Methods:

• Utilizing the Monte Carlo method in conjunction with fully decoupled belief propagation and ordered statistics decoding (FDBP-OSD).
• Developing a computational algorithm to determine the upper bound of code distance for arbitrary QSCs.
• Applying the algorithm to analyze Z-type Tanner-graph-recursive-expansion (Z-TGRE) codes and Chamon codes.

Main Results:

• The proposed FDBP-OSD algorithm accurately computes upper bounds for code distances, matching known values for various QSCs.
• For Z-TGRE codes, the computed upper bounds align with theoretical predictions.
• Analysis of Chamon codes suggests a potential code distance scaling of O(N^2/3), supporting existing conjectures.

Conclusions:

• The FDBP-OSD algorithm offers a precise and efficient method for calculating QSC code distance upper bounds.
• The findings provide valuable insights into the performance of Z-TGRE and Chamon codes.
• The results contribute to the understanding of code distance properties in complex quantum codes and support theoretical conjectures.