{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:44:24Z","timestamp":1760060664479,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T00:00:00Z","timestamp":1757376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Colleges and Universities Stable Support Project of Shenzhen, China","award":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"],"award-info":[{"award-number":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"]}]},{"name":"Shenzhen Science and Technology Program, China","award":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"],"award-info":[{"award-number":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"]}]},{"name":"Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies","award":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"],"award-info":[{"award-number":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"]}]},{"name":"Harbin Institute of Technology, Shenzhen\u2014SpinQ quantum information Joint Research Center Project","award":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"],"award-info":[{"award-number":["GXWD20220817164856008","GXWD20220811170225001","JCYJ20241202123906009","2022B1212010005","HITSZ20230111"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The code distance is a critical parameter of quantum stabilizer codes (QSCs), and determining it\u2014whether exactly or approximately\u2014is 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\u2014an 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.<\/jats:p>","DOI":"10.3390\/e27090940","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T12:04:55Z","timestamp":1757505895000},"page":"940","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation"],"prefix":"10.3390","volume":"27","author":[{"given":"Zhipeng","family":"Liang","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China"}]},{"given":"Zicheng","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China"}]},{"given":"Zhengzhong","family":"Yi","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"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"},{"name":"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"}]},{"given":"Fusheng","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China"}]},{"given":"Xuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,9]]},"reference":[{"key":"ref_1","unstructured":"Shor, P.W. 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