{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T17:20:31Z","timestamp":1763572831745,"version":"3.40.4"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"DOI":"10.1007\/s11128-025-04709-6","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T00:17:53Z","timestamp":1743121073000},"update-policy":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum stabilizer codes"],"prefix":"10.1007","volume":"24","author":[{"given":"Zhengzhong","family":"Yi","sequence":"first","affiliation":[]},{"given":"Zhipeng","family":"Liang","sequence":"additional","affiliation":[]},{"given":"Jiahan","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Kaixin","family":"Zhong","sequence":"additional","affiliation":[]},{"given":"Yulin","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Zhou","family":"Fang","sequence":"additional","affiliation":[]},{"given":"Xuan","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,25]]},"reference":[{"key":"4709_CR1","doi-asserted-by":"crossref","unstructured":"Campbell, E.: A series of fast-paced advances in quantum error correction. Nat. Rev. Phys. 1\u20132 (2024)","DOI":"10.1038\/s42254-024-00706-3"},{"key":"4709_CR2","doi-asserted-by":"publisher","first-page":"449","DOI":"10.4153\/CJM-1965-045-4","volume":"17","author":"J Edmonds","year":"1965","unstructured":"Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449\u2013467 (1965). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.4153\/CJM-1965-045-4","journal-title":"Can. J. Math."},{"issue":"3","key":"4709_CR3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.90.032326","volume":"90","author":"S Bravyi","year":"2014","unstructured":"Bravyi, S., Suchara, M., Vargo, A.: Efficient algorithms for maximum likelihood decoding in the surface code. Phys. Rev. A 90(3), 032326 (2014). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1103\/PhysRevA.90.032326","journal-title":"Phys. Rev. A"},{"issue":"5","key":"4709_CR4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.104.050504","volume":"104","author":"G Duclos-Cianci","year":"2010","unstructured":"Duclos-Cianci, G., Poulin, D.: Fast decoders for topological quantum codes. Phys. Rev. Lett. 104(5), 050504 (2010). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1103\/PhysRevLett.104.050504","journal-title":"Phys. Rev. Lett."},{"key":"4709_CR5","unstructured":"Schlingemann, D.: Stabilizer codes can be realized as graph codes. arXiv: quant-ph\/0111080 (2001)"},{"issue":"2","key":"4709_CR6","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1109\/18.748992","volume":"45","author":"DJ MacKay","year":"1999","unstructured":"MacKay, D.J.: Good error-correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory 45(2), 399\u2013431 (1999). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/18.748992","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4709_CR7","doi-asserted-by":"publisher","first-page":"2492","DOI":"10.1109\/ACCESS.2015.2503267","volume":"3","author":"Z Babar","year":"2015","unstructured":"Babar, Z., Botsinis, P., Alanis, D., Ng, S.X., Hanzo, L.: Fifteen years of quantum LDPC coding and improved decoding strategies. IEEE Access 3, 2492\u20132519 (2015). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/ACCESS.2015.2503267","journal-title":"IEEE Access"},{"issue":"1","key":"4709_CR8","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1038\/s41534-022-00623-2","volume":"8","author":"K-Y Kuo","year":"2022","unstructured":"Kuo, K.-Y., Lai, C.-Y.: Exploiting degeneracy in belief propagation decoding of quantum codes. npj Quantum Inf. 8(1), 111 (2022). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1038\/s41534-022-00623-2","journal-title":"npj Quantum Inf."},{"issue":"2","key":"4709_CR9","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1109\/JSAIT.2020.3011758","volume":"1","author":"K-Y Kuo","year":"2020","unstructured":"Kuo, K.-Y., Lai, C.-Y.: Refined belief propagation decoding of sparse-graph quantum codes. IEEE J. Sel. Areas Inf. Theory 1(2), 487\u2013498 (2020). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/JSAIT.2020.3011758","journal-title":"IEEE J. Sel. Areas Inf. Theory"},{"key":"4709_CR10","doi-asserted-by":"publisher","unstructured":"Kuo, K.-Y., Lai, C.-Y.: Refined belief-propagation decoding of quantum codes with scalar messages. In: 2020 IEEE Globecom Workshops (GC Wkshps), pp. 1\u20136. IEEE (2020). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/GCWkshps50303.2020.9367482","DOI":"10.1109\/GCWkshps50303.2020.9367482"},{"key":"4709_CR11","doi-asserted-by":"crossref","unstructured":"Poulin, D., Chung, Y.: On the iterative decoding of sparse quantum codes. Preprint at arXiv:0801.1241 (2008)","DOI":"10.26421\/QIC8.10-8"},{"issue":"4","key":"4709_CR12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.043423","volume":"2","author":"J Roffe","year":"2020","unstructured":"Roffe, J., White, D.R., Burton, S., Campbell, E.: Decoding across the quantum low-density parity-check code landscape. Phys. Rev. Res. 2(4), 043423 (2020). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1103\/PhysRevResearch.2.043423","journal-title":"Phys. Rev. Res."},{"key":"4709_CR13","doi-asserted-by":"publisher","first-page":"585","DOI":"10.22331\/q-2021-11-22-585","volume":"5","author":"P Panteleev","year":"2021","unstructured":"Panteleev, P., Kalachev, G.: Degenerate quantum LDPC codes with good finite length performance. Quantum 5, 585 (2021). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.22331\/q-2021-11-22-585","journal-title":"Quantum"},{"key":"4709_CR14","doi-asserted-by":"crossref","unstructured":"Kung, C.-F., Kuo, K.-Y., Lai, C.-Y.: On belief propagation decoding of quantum codes with quaternary reliability statistics. Preprint at arXiv:2305.03321 (2023)","DOI":"10.1109\/ISTC57237.2023.10273527"},{"key":"4709_CR15","doi-asserted-by":"crossref","unstructured":"Delfosse, N., Tillich, J.-P.: A decoding algorithm for CSS codes using the x\/z correlations. In: 2014 IEEE International Symposium on Information Theory, pp. 1071\u20131075. IEEE (2014)","DOI":"10.1109\/ISIT.2014.6874997"},{"key":"4709_CR16","unstructured":"Fowler, A.G.: Optimal complexity correction of correlated errors in the surface code. Preprint at arXiv:1310.0863 (2013)"},{"key":"4709_CR17","unstructured":"Yuan, Y., Lu, C.-C.: A modified MWPM decoding algorithm for quantum surface codes over depolarizing channels. Preprint at arXiv:2202.11239 (2022)"},{"key":"4709_CR18","doi-asserted-by":"publisher","unstructured":"Delfosse, N., Tillich, J.-P.: A decoding algorithm for CSS codes using the x\/z correlations. In: 2014 IEEE International Symposium on Information Theory, pp. 1071\u20131075 (2014). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/ISIT.2014.6874997","DOI":"10.1109\/ISIT.2014.6874997"},{"key":"4709_CR19","doi-asserted-by":"crossref","unstructured":"Li, J.X., Vontobel, P.O.: Lp decoding of quantum stabilizer codes. In: 2018 IEEE International Symposium on Information Theory (ISIT), pp. 1306\u20131310. IEEE (2018)","DOI":"10.1109\/ISIT.2018.8437905"},{"issue":"1","key":"4709_CR20","doi-asserted-by":"publisher","first-page":"2172","DOI":"10.1038\/s41467-021-22274-1","volume":"12","author":"JP Bonilla Ataides","year":"2021","unstructured":"Bonilla Ataides, J.P., Tuckett, D.K., Bartlett, S.D., Flammia, S.T., Brown, B.J.: The xzzx surface code. Nat. Commun. 12(1), 2172 (2021). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1038\/s41467-021-22274-1","journal-title":"Nat. Commun."},{"issue":"3","key":"4709_CR21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.86.032324","volume":"86","author":"AG Fowler","year":"2012","unstructured":"Fowler, A.G., Mariantoni, M., Martinis, J.M., Cleland, A.N.: Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86(3), 032324 (2012). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1103\/PhysRevA.86.032324","journal-title":"Phys. Rev. A"},{"issue":"5","key":"4709_CR22","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1109\/26.768759","volume":"47","author":"MP Fossorier","year":"1999","unstructured":"Fossorier, M.P., Mihaljevic, M., Imai, H.: Reduced complexity iterative decoding of low-density parity check codes based on belief propagation. IEEE Trans. Commun. 47(5), 673\u2013680 (1999). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/26.768759","journal-title":"IEEE Trans. Commun."},{"key":"4709_CR23","unstructured":"Roffe, J.: LDPC: Python Tools for Low Density Parity Check Codes. https:\/\/linproxy.fan.workers.dev:443\/https\/pypi.org\/project\/ldpc\/"},{"issue":"5","key":"4709_CR24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.83.052331","volume":"83","author":"M-H Hsieh","year":"2011","unstructured":"Hsieh, M.-H., Le Gall, F.: Np-hardness of decoding quantum error-correction codes. Phys. Rev. A 83(5), 052331 (2011). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1103\/PhysRevA.83.052331","journal-title":"Phys. Rev. A"},{"issue":"9","key":"4709_CR25","doi-asserted-by":"publisher","first-page":"5209","DOI":"10.1109\/TIT.2015.2422294","volume":"61","author":"P Iyer","year":"2015","unstructured":"Iyer, P., Poulin, D.: Hardness of decoding quantum stabilizer codes. IEEE Trans. Inf. Theory 61(9), 5209\u20135223 (2015). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/TIT.2015.2422294","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4709_CR26","unstructured":"Wiberg, N.: Codes and decoding on general graphs. PhD thesis, Department of electrical engineering, link\u00f6ping university Sweden (1996)"},{"issue":"2","key":"4709_CR27","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1109\/49.661103","volume":"16","author":"RJ McEliece","year":"1998","unstructured":"McEliece, R.J., MacKay, D.J.C., Cheng, J.-F.: Turbo decoding as an instance of pearl\u2019s \u201cbelief propagation\u2019\u2019 algorithm. IEEE J. Sel. Areas Commun. 16(2), 140\u2013152 (1998). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/49.661103","journal-title":"IEEE J. Sel. Areas Commun."},{"issue":"2","key":"4709_CR28","doi-asserted-by":"publisher","first-page":"498","DOI":"10.1109\/18.910572","volume":"47","author":"FR Kschischang","year":"2001","unstructured":"Kschischang, F.R., Frey, B.J., Loeliger, H.-A.: Factor graphs and the sum-product algorithm. IEEE Trans. Inf. Theory 47(2), 498\u2013519 (2001). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/18.910572","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4709_CR29","doi-asserted-by":"publisher","unstructured":"Emran, A.A., Elsabrouty, M.: Simplified variable-scaled min sum LDPC decoder for irregular LDPC codes. In: 2014 IEEE 11th Consumer Communications and Networking Conference (CCNC), pp. 518\u2013523. IEEE (2014). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/CCNC.2014.6940497","DOI":"10.1109\/CCNC.2014.6940497"},{"issue":"3","key":"4709_CR30","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1109\/26.990903","volume":"50","author":"J Chen","year":"2002","unstructured":"Chen, J., Fossorier, M.P.: Near optimum universal belief propagation based decoding of low-density parity check codes. IEEE Trans. Commun. 50(3), 406\u2013414 (2002). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/26.990903","journal-title":"IEEE Trans. Commun."},{"issue":"5","key":"4709_CR31","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1109\/4234.1001666","volume":"6","author":"J Chen","year":"2002","unstructured":"Chen, J., Fossorier, M.P.: Density evolution for two improved bp-based decoding algorithms of LDPC codes. IEEE Commun. Lett. 6(5), 208\u2013210 (2002). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/4234.1001666","journal-title":"IEEE Commun. Lett."},{"issue":"5","key":"4709_CR32","doi-asserted-by":"publisher","first-page":"1379","DOI":"10.1109\/18.412683","volume":"41","author":"MP Fossorier","year":"1995","unstructured":"Fossorier, M.P., Lin, S.: Soft-decision decoding of linear block codes based on ordered statistics. IEEE Trans. Inf. Theory 41(5), 1379\u20131396 (1995). https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1109\/18.412683","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s11128-025-04709-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/link.springer.com\/article\/10.1007\/s11128-025-04709-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s11128-025-04709-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T10:18:59Z","timestamp":1746440339000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/link.springer.com\/10.1007\/s11128-025-04709-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,25]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["4709"],"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.1007\/s11128-025-04709-6","relation":{},"ISSN":["1573-1332"],"issn-type":[{"type":"electronic","value":"1573-1332"}],"subject":[],"published":{"date-parts":[[2025,3,25]]},"assertion":[{"value":"29 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"Not applicable.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"Not applicable.","order":6,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}],"article-number":"100"}}