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Entropy
Volume 27
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10.3390/e27090940
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Article

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

by
Zhipeng Liang
1,
Zicheng Wang
1,
Zhengzhong Yi
2,3,4,*,
Fusheng Yang
1 and
Xuan Wang
1,*
1
School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China
2
Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
3
Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
4
Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China
*
Authors to whom correspondence should be addressed.
Entropy 2025, 27(9), 940; https://doi.org/10.3390/e27090940
Submission received: 18 July 2025 / Revised: 4 September 2025 / Accepted: 5 September 2025 / Published: 9 September 2025
(This article belongs to the Special Issue Quantum Error Correction and Fault-Tolerance)
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Abstract

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 X 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.
Keywords: quantum stabilizer code; code distance; quantum XYZ product code; Z-type Tanner-graph-recursive-expansion code; Chamon code quantum stabilizer code; code distance; quantum XYZ product code; Z-type Tanner-graph-recursive-expansion code; Chamon code

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MDPI and ACS Style

Liang, Z.; Wang, Z.; Yi, Z.; Yang, F.; Wang, X. Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation. Entropy 2025, 27, 940. https://doi.org/10.3390/e27090940

AMA Style

Liang Z, Wang Z, Yi Z, Yang F, Wang X. Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation. Entropy. 2025; 27(9):940. https://doi.org/10.3390/e27090940

Chicago/Turabian Style

Liang, Zhipeng, Zicheng Wang, Zhengzhong Yi, Fusheng Yang, and Xuan Wang. 2025. "Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation" Entropy 27, no. 9: 940. https://doi.org/10.3390/e27090940

APA Style

Liang, Z., Wang, Z., Yi, Z., Yang, F., & Wang, X. (2025). Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation. Entropy, 27(9), 940. https://doi.org/10.3390/e27090940

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