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Entanglement-assisted Quasi-cyclic Quantum Low-density Parity-check Codes over Qubits

Pavan Kumar, Abhi Kumar Sharma, Karthik Bharadwaj, and Shayan Srinivasa Garani

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AbstractWe construct several families of entanglement-assisted quasi-cyclic quantum LDPC (EA-QC-QLDPC) codes via structured tilings of permutation matrices. The entanglement-unassisted portion of the joint Tanner graph of the proposed EA-QC-QLDPC code derived from two distinct classical QC-LDPC codes is free of 4-cycles. Notably, one of the proposed families constructed from two distinct classical codes requires only a ${single}$ shared Bell pair between the quantum transmitter and receiver, highlighting its resource efficiency. We also analytically determine the exact code rates for some of the proposed constructions. Furthermore, two of the proposed families of EA-QC-QLDPC codes are derived from a single classical code whose Tanner graphs possess girth greater than six, further enhancing their error-correcting performance. We also propose an encoding scheme with improved complexity by exploiting the proposed code structure. The performance of the proposed codes is assessed under both random and burst error models under the depolarizing and Markovian noise actions. Simulation results reveal nearly one order of improvement in error-correction performance with the quaternary block-layered normalized min-sum (QBLNMS) decoder compared to the layered binary sum-product decoder over both depolarizing and Markovian channels. Using the QBLNMS decoder over a quaternary alphabet, we demonstrate that correlated Pauli errors can be effectively handled within the decoding framework. Furthermore, under the QBLNMS decoding, the proposed codes achieve ${significant}$ performance improvements compared to prior works and can effectively handle both random and burst errors. The code constructions are scalable across various coding rates and quantum payloads, crucial for practical quantum communication and computing systems.Featured image: In this scenario, we illustrate a two-way entanglement-assisted communication protocol from stations A and B that operate in catalysis mode. Each station thus acts as a transmitter as well as a receiver. This scenario is practical when the stations can do both computing and communication using a pool of Bell pairs. The transmitter has access to local Bell pair sources and one half of each Bell pair is encoded and sent from the transmitter to the receiver. At the receiver, the decoder uses the other half of the Bell pair assumed to be error free, decodes the information qubits as well as recovering the Bell pair at the receiver for consuming it in the next iteration. Thus, the Bell pairs are constantly cycling between the two stations, in a ping-pong fashion. The reader must note that one has to check for the validity of the decoded Bell pair for successfully using it in a subsequent round of transmission.Popular summaryQuantum information is extremely fragile and can be corrupted by noise arising from imperfect quantum hardware and interactions with the environment. Quantum error-correcting codes protect quantum information against these errors, but practical implementations require codes that not only provide strong error-correction capability but can also be encoded and decoded efficiently. Quantum low-density parity-check (LDPC) codes are promising candidates because of their sparse structure and scalability, yet many existing constructions suffer from short cycles in their joint Tanner graph that degrade performance, particularly when correcting correlated quantum errors. In this work, we construct several families of entanglement-assisted quasi-cyclic quantum LDPC (EA-QC-QLDPC) codes using structured tilings of permutation matrices. Our constructions produce scalable code families with analytically determined code parameters and, in one family, require only a single pre-shared Bell pair between the transmitter and receiver, making them highly resource efficient. We also design families whose underlying Tanner graphs have larger girth, which further improves decoding performance. Furthermore, when constructing entanglement-assisted QC-QLDPC codes from two different classical LDPC codes, we ensure that the entanglement-unassisted portion of the joint Tanner graph is free of 4-cycles, thereby improving the code's ability to handle correlated errors. To enable practical deployment, we develop an efficient encoding method that exploits structure of the proposed codes, substantially reducing encoding complexity compared with general-purpose stabilizer encoding techniques. We also employ a quaternary block-layered normalized min-sum (QBLNMS) decoder that operates on the entanglement-unassisted portion of joint Tanner graph, allowing correlated Pauli errors to be handled directly. Simulation results over both depolarizing and Markovian burst-error channels demonstrate that the proposed decoder achieves nearly an order-of-magnitude improvement in error-correction performance over layered binary sum-product decoding. Furthermore, the proposed code families significantly outperform existing quantum LDPC constructions. Compared with a previously reported quasi-cyclic CSS code of similar length, our codes achieve more than two orders-of-magnitude lower error rates by eliminating 4-cycles in the entanglement-unassisted portion of joint Tanner graph. Compared with a prior entanglement-assisted quantum LDPC construction, the proposed codes provide more than an order-of-magnitude improvement under depolarizing noise. We also demonstrate improved resilience to burst errors and show that larger Tanner graph girth and increased column weight further enhance decoding performance. These results show that carefully designed structured entanglement-assisted quantum LDPC codes can simultaneously improve reliability, reduce implementation complexity, and lower entanglement requirements. The proposed constructions therefore represent a promising step toward practical, high-performance quantum error correction for future quantum communication networks and fault-tolerant quantum computing.► BibTeX data@article{Kumar2026entanglement, doi = {10.22331/q-2026-07-31-2181}, url = {https://doi.org/10.22331/q-2026-07-31-2181}, title = {Entanglement-assisted {Q}uasi-cyclic {Q}uantum {L}ow-density {P}arity-check {C}odes over {Q}ubits}, author = {Kumar, Pavan and Sharma, Abhi Kumar and Bharadwaj, Karthik and Garani, Shayan Srinivasa}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2181}, month = jul, year = {2026} }► References [1] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. California Institute of Technology. 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Quantum 8, 1420 (2024). https://doi.org/10.22331/q-2024-07-22-1420Cited byCould not fetch Crossref cited-by data during last attempt 2026-07-31 08:52:25: Could not fetch cited-by data for 10.22331/q-2026-07-31-2181 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-07-31 08:52:25: Cannot retrieve data from ADS due to rate limitations.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. AbstractWe construct several families of entanglement-assisted quasi-cyclic quantum LDPC (EA-QC-QLDPC) codes via structured tilings of permutation matrices. The entanglement-unassisted portion of the joint Tanner graph of the proposed EA-QC-QLDPC code derived from two distinct classical QC-LDPC codes is free of 4-cycles. Notably, one of the proposed families constructed from two distinct classical codes requires only a ${single}$ shared Bell pair between the quantum transmitter and receiver, highlighting its resource efficiency. We also analytically determine the exact code rates for some of the proposed constructions. Furthermore, two of the proposed families of EA-QC-QLDPC codes are derived from a single classical code whose Tanner graphs possess girth greater than six, further enhancing their error-correcting performance. We also propose an encoding scheme with improved complexity by exploiting the proposed code structure. The performance of the proposed codes is assessed under both random and burst error models under the depolarizing and Markovian noise actions. Simulation results reveal nearly one order of improvement in error-correction performance with the quaternary block-layered normalized min-sum (QBLNMS) decoder compared to the layered binary sum-product decoder over both depolarizing and Markovian channels. Using the QBLNMS decoder over a quaternary alphabet, we demonstrate that correlated Pauli errors can be effectively handled within the decoding framework. Furthermore, under the QBLNMS decoding, the proposed codes achieve ${significant}$ performance improvements compared to prior works and can effectively handle both random and burst errors. The code constructions are scalable across various coding rates and quantum payloads, crucial for practical quantum communication and computing systems.Featured image: In this scenario, we illustrate a two-way entanglement-assisted communication protocol from stations A and B that operate in catalysis mode. Each station thus acts as a transmitter as well as a receiver. This scenario is practical when the stations can do both computing and communication using a pool of Bell pairs. The transmitter has access to local Bell pair sources and one half of each Bell pair is encoded and sent from the transmitter to the receiver. At the receiver, the decoder uses the other half of the Bell pair assumed to be error free, decodes the information qubits as well as recovering the Bell pair at the receiver for consuming it in the next iteration. Thus, the Bell pairs are constantly cycling between the two stations, in a ping-pong fashion. The reader must note that one has to check for the validity of the decoded Bell pair for successfully using it in a subsequent round of transmission.Popular summaryQuantum information is extremely fragile and can be corrupted by noise arising from imperfect quantum hardware and interactions with the environment. Quantum error-correcting codes protect quantum information against these errors, but practical implementations require codes that not only provide strong error-correction capability but can also be encoded and decoded efficiently. Quantum low-density parity-check (LDPC) codes are promising candidates because of their sparse structure and scalability, yet many existing constructions suffer from short cycles in their joint Tanner graph that degrade performance, particularly when correcting correlated quantum errors. In this work, we construct several families of entanglement-assisted quasi-cyclic quantum LDPC (EA-QC-QLDPC) codes using structured tilings of permutation matrices. Our constructions produce scalable code families with analytically determined code parameters and, in one family, require only a single pre-shared Bell pair between the transmitter and receiver, making them highly resource efficient. We also design families whose underlying Tanner graphs have larger girth, which further improves decoding performance. Furthermore, when constructing entanglement-assisted QC-QLDPC codes from two different classical LDPC codes, we ensure that the entanglement-unassisted portion of the joint Tanner graph is free of 4-cycles, thereby improving the code's ability to handle correlated errors. To enable practical deployment, we develop an efficient encoding method that exploits structure of the proposed codes, substantially reducing encoding complexity compared with general-purpose stabilizer encoding techniques. We also employ a quaternary block-layered normalized min-sum (QBLNMS) decoder that operates on the entanglement-unassisted portion of joint Tanner graph, allowing correlated Pauli errors to be handled directly. Simulation results over both depolarizing and Markovian burst-error channels demonstrate that the proposed decoder achieves nearly an order-of-magnitude improvement in error-correction performance over layered binary sum-product decoding. Furthermore, the proposed code families significantly outperform existing quantum LDPC constructions. Compared with a previously reported quasi-cyclic CSS code of similar length, our codes achieve more than two orders-of-magnitude lower error rates by eliminating 4-cycles in the entanglement-unassisted portion of joint Tanner graph. Compared with a prior entanglement-assisted quantum LDPC construction, the proposed codes provide more than an order-of-magnitude improvement under depolarizing noise. We also demonstrate improved resilience to burst errors and show that larger Tanner graph girth and increased column weight further enhance decoding performance. These results show that carefully designed structured entanglement-assisted quantum LDPC codes can simultaneously improve reliability, reduce implementation complexity, and lower entanglement requirements. The proposed constructions therefore represent a promising step toward practical, high-performance quantum error correction for future quantum communication networks and fault-tolerant quantum computing.► BibTeX data@article{Kumar2026entanglement, doi = {10.22331/q-2026-07-31-2181}, url = {https://doi.org/10.22331/q-2026-07-31-2181}, title = {Entanglement-assisted {Q}uasi-cyclic {Q}uantum {L}ow-density {P}arity-check {C}odes over {Q}ubits}, author = {Kumar, Pavan and Sharma, Abhi Kumar and Bharadwaj, Karthik and Garani, Shayan Srinivasa}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2181}, month = jul, year = {2026} }► References [1] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. California Institute of Technology. (1997). arXiv:quant-ph/9705052 [2] A Robert Calderbank, Eric M Rains, Peter W Shor, and Neil JA Sloane. ``Quantum error correction and orthogonal geometry''. Phys. Rev. Lett. 78, 405 (1997). https://doi.org/10.1103/PhysRevLett.78.405 [3] A Robert Calderbank, Eric M Rains, Peter M Shor, and Neil JA Sloane. ``Quantum error correction via codes over GF (4)''. IEEE Trans. Inf. Theory 44, 1369–1387 (1998). https://doi.org/10.1109/18.681315 [4] Daniel A. Lidar and Todd A. Brun. ``Quantum error correction''.

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Entanglement-assisted Quasi-cyclic Quantum Low-density Parity-check Codes over Qubits

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