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Qudit low-density parity-check codes

Quantum Journal

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AbstractQudits offer significant advantages over qubit-based architectures, including more efficient gate compilation, reduced resource requirements, improved error-correction primitives, and enhanced capabilities for quantum communication and cryptography. 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.Featured image: Tanner graph representing an LDPC code, where instead of the circles representing qubits, they represent qudits, as indicated by the inset. The squares represent the X- and Z-type checks.Popular summaryError correction is believed to be essential for the realization of large-scale, fault-tolerant quantum computers. One of the most promising families of quantum error correcting codes is known as low-density parity-check (LDPC) codes, so-named for the small number of qubits involved in each stabilizer check measurement. LDPC codes have seen rapid development over the last few years, but most of this progress has been for qubit-based systems. In this work, we develop a unifying framework for qudit LDPC codes, where qudits generalize qubits by allowing quantum systems with more than two basis states. Qudits allow for a larger space for computation and admit several benefits over qubits, such as lower circuit complexity. We generalize several promising LDPC codes from qubits to qudits, such as the popular bivariate bicycle codes and hypergraph product codes, and we find several novel qudit codes that could be implemented on near-term qudit devices. Finally, this work serves as a reference for researchers working in qudit error correction, where we have brought together several mathematical concepts, various code constructions, and qudit formalism to give a comprehensive guide on qudit LDPC error correcting codes.► BibTeX data@article{Spencer2026quditlowdensity, doi = {10.22331/q-2026-03-13-2023}, url = {https://doi.org/10.22331/q-2026-03-13-2023}, title = {Qudit low-density parity-check codes}, author = {Spencer, Daniel J. and Tanggara, Andrew and Haug, Tobias and Khu, Derek and Bharti, Kishor}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2023}, month = mar, year = {2026} }► References [1] Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. 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Physical Review Letters, 122 (23): 230501, 2019. https://doi.org/10.1103/PhysRevLett.122.230501. https://doi.org/10.1103/PhysRevLett.122.230501 [109] Weilei Zeng and Leonid P Pryadko. Minimal distances for certain quantum product codes and tensor products of chain complexes. Physical Review A, 102 (6): 062402, 2020. https://doi.org/10.1103/PhysRevA.102.062402. https://doi.org/10.1103/PhysRevA.102.062402Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-13 11:27:41: Could not fetch cited-by data for 10.22331/q-2026-03-13-2023 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-13 11:27:41: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractQudits offer significant advantages over qubit-based architectures, including more efficient gate compilation, reduced resource requirements, improved error-correction primitives, and enhanced capabilities for quantum communication and cryptography. 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.Featured image: Tanner graph representing an LDPC code, where instead of the circles representing qubits, they represent qudits, as indicated by the inset. The squares represent the X- and Z-type checks.Popular summaryError correction is believed to be essential for the realization of large-scale, fault-tolerant quantum computers. One of the most promising families of quantum error correcting codes is known as low-density parity-check (LDPC) codes, so-named for the small number of qubits involved in each stabilizer check measurement. LDPC codes have seen rapid development over the last few years, but most of this progress has been for qubit-based systems. In this work, we develop a unifying framework for qudit LDPC codes, where qudits generalize qubits by allowing quantum systems with more than two basis states. Qudits allow for a larger space for computation and admit several benefits over qubits, such as lower circuit complexity. We generalize several promising LDPC codes from qubits to qudits, such as the popular bivariate bicycle codes and hypergraph product codes, and we find several novel qudit codes that could be implemented on near-term qudit devices. Finally, this work serves as a reference for researchers working in qudit error correction, where we have brought together several mathematical concepts, various code constructions, and qudit formalism to give a comprehensive guide on qudit LDPC error correcting codes.► BibTeX data@article{Spencer2026quditlowdensity, doi = {10.22331/q-2026-03-13-2023}, url = {https://doi.org/10.22331/q-2026-03-13-2023}, title = {Qudit low-density parity-check codes}, author = {Spencer, Daniel J. and Tanggara, Andrew and Haug, Tobias and Khu, Derek and Bharti, Kishor}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2023}, month = mar, year = {2026} }► References [1] Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. 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