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Yet, one of the most promising families of quantum error correction codes, namely quantum low-density parity-check (LDPC) codes, have so far been mostly restricted to qubits. Here, we generalize recent advancements in LDPC codes from qubits to qudits. We introduce a general framework for finding qudit LDPC codes and apply our formalism to several promising types of LDPC codes. We generalize bivariate bicycle codes, including their coprime variant; hypergraph product codes, including the recently proposed La-cross codes; subsystem hypergraph product (SHYPS) codes; high-dimensional expander codes, which make use of Ramanujan complexes; and fiber bundle codes. Using the qudit generalization formalism, we then numerically search for and decode several novel qudit codes compatible with near-term hardware. Our results highlight the potential of qudit LDPC codes as a versatile and hardware-compatible pathway toward scalable quantum error correction.<\/jats:p>","DOI":"10.22331\/q-2026-03-13-2023","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T11:32:33Z","timestamp":1773401553000},"page":"2023","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Qudit low-density parity-check codes"],"prefix":"10.22331","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1547-2935","authenticated-orcid":false,"given":"Daniel J.","family":"Spencer","sequence":"first","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD 20742, USA"},{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD 20742, USA"},{"name":"Department of Physics, University of Maryland, College Park, MD 20742, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrew","family":"Tanggara","sequence":"additional","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543"},{"name":"Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639673"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2707-9962","authenticated-orcid":false,"given":"Tobias","family":"Haug","sequence":"additional","affiliation":[{"name":"Quantum Research Center, Technology Innovation Institute, Abu Dhabi, UAE"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Derek","family":"Khu","sequence":"additional","affiliation":[{"name":"Institute for Infocomm Research (I2R), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #21-01, Connexis South Tower, Singapore 138632, Republic of Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7776-6608","authenticated-orcid":false,"given":"Kishor","family":"Bharti","sequence":"additional","affiliation":[{"name":"Quantum Innovation Centre (Q.InC), Agency for Science, Technology and Research (A*STAR), 2 Fusionopolis Way, Innovis #08-03, Singapore 138634, Republic of Singapore"},{"name":"Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Republic of Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2026,3,13]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Google Quantum AI. 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Annals of Physics, 303: 2\u201330, 2003. https:\/\/doi.org\/10.1016\/S0003-4916(02)00018-0.","DOI":"10.1016\/S0003-4916(02)00018-0"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Alexey A. Kovalev and Leonid P. Pryadko. Quantum Kronecker sum-product low-density parity-check codes with finite rate. Physical Review A, 88: 012311, 2013. https:\/\/doi.org\/10.1103\/PhysRevA.88.012311.","DOI":"10.1103\/PhysRevA.88.012311"},{"key":"56","doi-asserted-by":"publisher","unstructured":"Saunders Mac Lane. Homology. Springer Berlin, Heidelberg, 1 edition, 1995. ISBN 978-3-642-62029-4. https:\/\/doi.org\/10.1007\/978-3-642-62029-4.","DOI":"10.1007\/978-3-642-62029-4"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Florian M Leupold, Maciej Malinowski, Chi Zhang, Vlad Negnevitsky, Ad\u00e1n Cabello, Joseba Alonso, and Jonathan P Home. Sustained state-independent quantum contextual correlations from a single ion. 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MacKay and Radford M. Neal. Near Shannon limit performance of low density parity check codes. Electronics letters, 32, 1996. https:\/\/doi.org\/10.1049\/el:19961141.","DOI":"10.1049\/el:19961141"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Alexander J. Malcolm, Andrew N. Glaudell, Patricio Fuentes, Daryus Chandra, Alexis Schotte, Colby DeLisle, Rafael Haenel, Amir Ebrahimi, Joschka Roffe, Armanda O. Quintavalle, Stefanie J. Beale, Nicholas R. Lee-Hone, and Stephanie Simmons. Computing efficiently in QLDPC codes. arXiv:2502.07150, 2025. https:\/\/doi.org\/10.48550\/arXiv.2502.07150.","DOI":"10.48550\/arXiv.2502.07150"},{"key":"72","doi-asserted-by":"publisher","unstructured":"Michael Meth, Jan F Haase, Jinglei Zhang, Claire Edmunds, Lukas Postler, Alex Steiner, Andrew J Jena, Luca Dellantonio, Rainer Blatt, Peter Zoller, et al. Simulating 2d lattice gauge theories on a qudit quantum computer. 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Quantum, 5: 585, 2021a. https:\/\/doi.org\/10.22331\/q-2021-11-22-585.","DOI":"10.22331\/q-2021-11-22-585"},{"key":"81","doi-asserted-by":"publisher","unstructured":"Pavel Panteleev and Gleb Kalachev. Degenerate quantum LDPC codes with good finite length performance. Quantum, 5: 585, 2021b. https:\/\/doi.org\/10.22331\/q-2021-11-22-585.","DOI":"10.22331\/q-2021-11-22-585"},{"key":"82","doi-asserted-by":"publisher","unstructured":"Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical LDPC codes. STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 375\u2013388, 2022. https:\/\/doi.org\/10.1145\/3519935.3520017.","DOI":"10.1145\/3519935.3520017"},{"key":"83","doi-asserted-by":"publisher","unstructured":"Laura Pecorari, Sven Jandura, Gavin K. Brennen, and Guido Pupillo. High-rate quantum LDPC codes for long-range-connected neutral atom registers. 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Proceedings of the Royal Society A, 452, 1996. https:\/\/doi.org\/10.1098\/rspa.1996.0136.","DOI":"10.1098\/rspa.1996.0136"},{"key":"95","doi-asserted-by":"publisher","unstructured":"Fracesco Tacchino, Alessandro Chiesa, Roberta Sessoli, Ivano Tavernelli, and Stefano Carretta. A proposal for using molecular spin qudits as quantum simulators of light\u2013matter interactions. Journal of Materials Chemistry C, 9 (32): 10266\u201310275, 2021. https:\/\/doi.org\/10.1039\/D1TC00851J.","DOI":"10.1039\/D1TC00851J"},{"key":"96","doi-asserted-by":"publisher","unstructured":"Jean-Pierre Tillich and Gilles Zemor. Quantum LDPC codes with positive rate and minimum distance proportional to square root of the blocklength. IEEE Transactions on Information Theory, 60, 2014. https:\/\/doi.org\/10.1109\/TIT.2013.2292061.","DOI":"10.1109\/TIT.2013.2292061"},{"key":"97","doi-asserted-by":"publisher","unstructured":"Giacomo Torlai and Roger G Melko. Neural decoder for topological codes. 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Yoder, Eddie Schoute, Patrick Rall, Emily Pritchett, Jay M. Gambetta, Andrew W. Cross, Malcolm Carroll, and Michael E. Beverland. Tour de gross: A modular quantum computer based on bivariate bicycle codes. arXiv:2506.03094, 2025. https:\/\/doi.org\/10.48550\/arXiv.2506.03094.","DOI":"10.48550\/arXiv.2506.03094"},{"key":"107","doi-asserted-by":"publisher","unstructured":"Weilei Zeng and Leonid P Pryadko. Higher-dimensional quantum hypergraph-product codes with finite rates. Physical Review Letters, 122 (23): 230501, 2019. https:\/\/doi.org\/10.1103\/PhysRevLett.122.230501.","DOI":"10.1103\/PhysRevLett.122.230501"},{"key":"108","doi-asserted-by":"publisher","unstructured":"Weilei Zeng and Leonid P Pryadko. Minimal distances for certain quantum product codes and tensor products of chain complexes. 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