Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms
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AbstractQuantum computing is deemed to require error correction at scale to mitigate physical noise by reducing it to lower noise levels while operating on encoded logical qubits. Popular quantum error correction schemes include CSS code, of which surface codes provide regular mappings onto 2D planes suitable for contemporary quantum devices together with known transversal logical gates. Recently, qLDPC codes have been proposed as a means to provide denser encoding with the class of bivariate bicycle (BB) codes promising feasible design for devices. This work contributes a novel subclass of BB codes suitable for quantum error correction. This subclass employs $coprimes$ and the product $xy$ of the two generating variables $x$ and $y$ to construct polynomials, rather than using $x$ and $y$ separately as in vanilla BB codes. In contrast to vanilla BB codes, where parameters remain unknown prior to code discovery, the rate of the proposed code can be determined beforehand by specifying a factor polynomial as an input to the numerical search algorithm. Using this coprime-BB construction, we found a number of surprisingly short to medium-length codes that were previously unknown. We also propose a layout on cold atom arrays tailored for coprime-BB codes. The proposed layout reduces both move time for short to medium-length codes and the number of moves of atoms to perform syndrome extractions. We consider an error model with global laser noise on cold atoms, and simulations show that our proposed layout achieves significant improvements over prior work across the simulated codes.► BibTeX data@article{Wang2026coprimebivariate, doi = {10.22331/q-2026-02-23-2009}, url = {https://doi.org/10.22331/q-2026-02-23-2009}, title = {Coprime {B}ivariate {B}icycle {C}odes and {T}heir {L}ayouts on {C}old {A}toms}, author = {Wang, Ming and Mueller, Frank}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2009}, month = feb, year = {2026} }► References [1] D.J.C. MacKay, G. Mitchison, and P.L. McFadden. ``Sparse-graph codes for quantum error correction''. IEEE Transactions on Information Theory 50, 2315–2330 (2004). Appearances:. https://doi.org/10.1109/TIT.2004.834737 [2] Jean-Pierre Tillich and Gilles Zémor. ``Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength''. IEEE Transactions on Information Theory 60, 1193–1202 (2014). Appearances:. https://doi.org/10.1109/TIT.2013.2292061 [3] Pavel Panteleev and Gleb Kalachev. ``Asymptotically good quantum and locally testable classical LDPC codes''. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Page 375–388. STOC 2022New York, NY, USA (2022). Association for Computing Machinery. Appearances:. https://doi.org/10.1145/3519935.3520017 [4] Nikolas P. Breuckmann and Jens Niklas Eberhardt. ``Quantum low-density parity-check codes''. PRX Quantum 2, 040101 (2021). Appearances:. https://doi.org/10.1103/PRXQuantum.2.040101 [5] Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. ``High-threshold and low-overhead fault-tolerant quantum memory''. Nature 627, 778–782 (2024). Appearances:. https://doi.org/10.1038/s41586-024-07107-7 [6] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Phys. Rev. A 86, 032324 (2012). Appearances:. https://doi.org/10.1103/PhysRevA.86.032324 [7] Pavel Panteleev and Gleb Kalachev. ``Degenerate quantum LDPC codes with good finite length performance''. Quantum 5, 585 (2021). Appearances:. https://doi.org/10.22331/q-2021-11-22-585 [8] Joshua Viszlai, Willers Yang, Sophia Fuhui Lin, Junyu Liu, Natalia Nottingham, Jonathan M Baker, and Frederic T Chong. ``Matching generalized-bicycle codes to neutral atoms for low-overhead fault-tolerance''. In 2025 IEEE International Conference on Quantum Computing and Engineering (QCE). Volume 01, pages 688–699. (2025). Appearances:. https://doi.org/10.1109/QCE65121.2025.00080 [9] C. Poole, T. M. Graham, M. A. Perlin, M. Otten, and M. Saffman. ``Architecture for fast implementation of quantum low-density parity-check codes with optimized rydberg gates''. Phys. Rev. A 111, 022433 (2025). Appearances:. https://doi.org/10.1103/PhysRevA.111.022433 [10] Yifan Hong, Elijah Durso-Sabina, David Hayes, and Andrew Lucas. ``Entangling four logical qubits beyond break-even in a nonlocal code''. Phys. Rev. Lett. 133, 180601 (2024). Appearances:. https://doi.org/10.1103/PhysRevLett.133.180601 [11] Noah Berthusen, Dhruv Devulapalli, Eddie Schoute, Andrew M. Childs, Michael J. Gullans, Alexey V. Gorshkov, and Daniel Gottesman. ``Toward a 2d local implementation of quantum low-density parity-check codes''. PRX Quantum 6, 010306 (2025). Appearances:. https://doi.org/10.1103/PRXQuantum.6.010306 [12] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098–1105 (1996). Appearances:. https://doi.org/10.1103/PhysRevA.54.1098 [13] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. ``Logical quantum processor based on reconfigurable atom arrays''. Nature 626, 58–65 (2024). Appearances:. https://doi.org/10.1038/s41586-023-06927-3 [14] Bochen Tan, Dolev Bluvstein, Mikhail D. Lukin, and Jason Cong. ``Qubit mapping for reconfigurable atom arrays''. In 2022 IEEE/ACM International Conference On Computer Aided Design (ICCAD). Pages 1–9. (2022). [15] Hsiang-Ku Lin and Leonid P. Pryadko. ``Quantum two-block group algebra codes''. Phys. Rev. A 109, 022407 (2024). Appearances:. https://doi.org/10.1103/PhysRevA.109.022407 [16] Jens Niklas Eberhardt and Vincent Steffan. ``Logical operators and fold-transversal gates of bivariate bicycle codes''. IEEE Transactions on Information Theory 71, 1140–1152 (2025). Appearances:. https://doi.org/10.1109/TIT.2024.3521638 [17] Qian Xu, J Pablo Bonilla Ataides, Christopher A Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D Lukin, Liang Jiang, and Hengyun Zhou. ``Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays''. Nature Physics 20, 1084–1090 (2024). Appearances:. https://doi.org/10.1038/s41567-024-02479-z [18] Jasper Johannes Postema and Servaas JJMF Kokkelmans. ``Existence and characterisation of coprime bivariate bicycle codes'' (2025). [19] Neng-Chun Chiu, Elias C. Trapp, Jinen Guo, Mohamed H. Abobeih, Luke M. Stewart, Simon Hollerith, Pavel L. Stroganov, Marcin Kalinowski, Alexandra A. Geim, Simon J. Evered, Sophie H. Li, Xingjian Lyu, Lisa M. Peters, Dolev Bluvstein, Tout T. Wang, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``Continuous operation of a coherent 3,000-qubit system''. Nature 646, 1075–1080 (2025). Appearances:. https://doi.org/10.1038/s41586-025-09596-6 [20] Shuai Wang, Wenjun Zhang, Tao Zhang, Shuyao Mei, Yuqing Wang, Jiazhong Hu, and Wenlan Chen. ``Accelerating the assembly of defect-free atomic arrays with maximum parallelisms''. Phys. Rev. Appl. 19, 054032 (2023). Appearances:. https://doi.org/10.1103/PhysRevApplied.19.054032 [21] K. O. Roberts, T. McKellar, J. Fekete, A. Rakonjac, A. B. Deb, and N. Kjærgaard. ``Steerable optical tweezers for ultracold atom studies''. Opt. Lett. 39, 2012–2015 (2014). Appearances:. https://doi.org/10.1364/OL.39.002012 [22] Craig Gidney. ``Stim: a fast stabilizer circuit simulator''. Quantum 5, 497 (2021). Appearances:. https://doi.org/10.22331/q-2021-07-06-497 [23] Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. ``Decoding across the quantum low-density parity-check code landscape''. Phys. Rev. Res. 2, 043423 (2020). Appearances:. https://doi.org/10.1103/PhysRevResearch.2.043423 [24] Nikolaos Koukoulekidis, Fedor Šimkovic IV, Martin Leib, and Francisco Revson Fernandes Pereira. ``Small quantum codes from algebraic extensions of generalized bicycle codes'' (2024). arXiv:2401.07583. Appearances:. arXiv:2401.07583 [25] Lukas Voss, Sim Jian Xian, Tobias Haug, and Kishor Bharti. ``Multivariate bicycle codes''. Phys. Rev. A 111, L060401 (2025). Appearances:. https://doi.org/10.1103/ll5p-z88p [26] Mackenzie H. Shaw and Barbara M. Terhal. ``Lowering connectivity requirements for bivariate bicycle codes using morphing circuits''. Phys. Rev. Lett. 134, 090602 (2025). Appearances:. https://doi.org/10.1103/PhysRevLett.134.090602Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-23 13:36:47: Could not fetch cited-by data for 10.22331/q-2026-02-23-2009 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-23 13:36:47: 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. AbstractQuantum computing is deemed to require error correction at scale to mitigate physical noise by reducing it to lower noise levels while operating on encoded logical qubits. Popular quantum error correction schemes include CSS code, of which surface codes provide regular mappings onto 2D planes suitable for contemporary quantum devices together with known transversal logical gates. Recently, qLDPC codes have been proposed as a means to provide denser encoding with the class of bivariate bicycle (BB) codes promising feasible design for devices. This work contributes a novel subclass of BB codes suitable for quantum error correction. This subclass employs $coprimes$ and the product $xy$ of the two generating variables $x$ and $y$ to construct polynomials, rather than using $x$ and $y$ separately as in vanilla BB codes. In contrast to vanilla BB codes, where parameters remain unknown prior to code discovery, the rate of the proposed code can be determined beforehand by specifying a factor polynomial as an input to the numerical search algorithm. Using this coprime-BB construction, we found a number of surprisingly short to medium-length codes that were previously unknown. We also propose a layout on cold atom arrays tailored for coprime-BB codes. The proposed layout reduces both move time for short to medium-length codes and the number of moves of atoms to perform syndrome extractions. We consider an error model with global laser noise on cold atoms, and simulations show that our proposed layout achieves significant improvements over prior work across the simulated codes.► BibTeX data@article{Wang2026coprimebivariate, doi = {10.22331/q-2026-02-23-2009}, url = {https://doi.org/10.22331/q-2026-02-23-2009}, title = {Coprime {B}ivariate {B}icycle {C}odes and {T}heir {L}ayouts on {C}old {A}toms}, author = {Wang, Ming and Mueller, Frank}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2009}, month = feb, year = {2026} }► References [1] D.J.C. MacKay, G. Mitchison, and P.L. McFadden. ``Sparse-graph codes for quantum error correction''. IEEE Transactions on Information Theory 50, 2315–2330 (2004). Appearances:. https://doi.org/10.1109/TIT.2004.834737 [2] Jean-Pierre Tillich and Gilles Zémor. ``Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength''. IEEE Transactions on Information Theory 60, 1193–1202 (2014). Appearances:. https://doi.org/10.1109/TIT.2013.2292061 [3] Pavel Panteleev and Gleb Kalachev. ``Asymptotically good quantum and locally testable classical LDPC codes''. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Page 375–388. STOC 2022New York, NY, USA (2022). Association for Computing Machinery. Appearances:. https://doi.org/10.1145/3519935.3520017 [4] Nikolas P. Breuckmann and Jens Niklas Eberhardt. ``Quantum low-density parity-check codes''. PRX Quantum 2, 040101 (2021). Appearances:. https://doi.org/10.1103/PRXQuantum.2.040101 [5] Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder. ``High-threshold and low-overhead fault-tolerant quantum memory''. Nature 627, 778–782 (2024). Appearances:. https://doi.org/10.1038/s41586-024-07107-7 [6] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Phys. Rev. A 86, 032324 (2012). Appearances:. https://doi.org/10.1103/PhysRevA.86.032324 [7] Pavel Panteleev and Gleb Kalachev. ``Degenerate quantum LDPC codes with good finite length performance''. Quantum 5, 585 (2021). Appearances:. https://doi.org/10.22331/q-2021-11-22-585 [8] Joshua Viszlai, Willers Yang, Sophia Fuhui Lin, Junyu Liu, Natalia Nottingham, Jonathan M Baker, and Frederic T Chong. ``Matching generalized-bicycle codes to neutral atoms for low-overhead fault-tolerance''. In 2025 IEEE International Conference on Quantum Computing and Engineering (QCE). Volume 01, pages 688–699. (2025). Appearances:. https://doi.org/10.1109/QCE65121.2025.00080 [9] C. Poole, T. M. Graham, M. A. Perlin, M. Otten, and M. Saffman. ``Architecture for fast implementation of quantum low-density parity-check codes with optimized rydberg gates''. Phys. Rev. A 111, 022433 (2025). Appearances:. https://doi.org/10.1103/PhysRevA.111.022433 [10] Yifan Hong, Elijah Durso-Sabina, David Hayes, and Andrew Lucas. ``Entangling four logical qubits beyond break-even in a nonlocal code''. Phys. Rev. Lett. 133, 180601 (2024). Appearances:. https://doi.org/10.1103/PhysRevLett.133.180601 [11] Noah Berthusen, Dhruv Devulapalli, Eddie Schoute, Andrew M. Childs, Michael J. Gullans, Alexey V. Gorshkov, and Daniel Gottesman. ``Toward a 2d local implementation of quantum low-density parity-check codes''. PRX Quantum 6, 010306 (2025). Appearances:. https://doi.org/10.1103/PRXQuantum.6.010306 [12] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098–1105 (1996). Appearances:. https://doi.org/10.1103/PhysRevA.54.1098 [13] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. ``Logical quantum processor based on reconfigurable atom arrays''. Nature 626, 58–65 (2024). Appearances:. https://doi.org/10.1038/s41586-023-06927-3 [14] Bochen Tan, Dolev Bluvstein, Mikhail D. Lukin, and Jason Cong. ``Qubit mapping for reconfigurable atom arrays''. In 2022 IEEE/ACM International Conference On Computer Aided Design (ICCAD). Pages 1–9. (2022). [15] Hsiang-Ku Lin and Leonid P. Pryadko. ``Quantum two-block group algebra codes''. Phys. Rev. A 109, 022407 (2024). Appearances:. https://doi.org/10.1103/PhysRevA.109.022407 [16] Jens Niklas Eberhardt and Vincent Steffan. ``Logical operators and fold-transversal gates of bivariate bicycle codes''. IEEE Transactions on Information Theory 71, 1140–1152 (2025). Appearances:. https://doi.org/10.1109/TIT.2024.3521638 [17] Qian Xu, J Pablo Bonilla Ataides, Christopher A Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasić, Mikhail D Lukin, Liang Jiang, and Hengyun Zhou. ``Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays''. Nature Physics 20, 1084–1090 (2024). Appearances:. https://doi.org/10.1038/s41567-024-02479-z [18] Jasper Johannes Postema and Servaas JJMF Kokkelmans. ``Existence and characterisation of coprime bivariate bicycle codes'' (2025). [19] Neng-Chun Chiu, Elias C. Trapp, Jinen Guo, Mohamed H. Abobeih, Luke M. Stewart, Simon Hollerith, Pavel L. Stroganov, Marcin Kalinowski, Alexandra A. Geim, Simon J. Evered, Sophie H. Li, Xingjian Lyu, Lisa M. Peters, Dolev Bluvstein, Tout T. Wang, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. ``Continuous operation of a coherent 3,000-qubit system''. Nature 646, 1075–1080 (2025). Appearances:. https://doi.org/10.1038/s41586-025-09596-6 [20] Shuai Wang, Wenjun Zhang, Tao Zhang, Shuyao Mei, Yuqing Wang, Jiazhong Hu, and Wenlan Chen. ``Accelerating the assembly of defect-free atomic arrays with maximum parallelisms''. Phys. Rev. Appl. 19, 054032 (2023). Appearances:. https://doi.org/10.1103/PhysRevApplied.19.054032 [21] K. O. Roberts, T. McKellar, J. Fekete, A. Rakonjac, A. B. Deb, and N. Kjærgaard. ``Steerable optical tweezers for ultracold atom studies''. Opt. Lett. 39, 2012–2015 (2014). Appearances:. https://doi.org/10.1364/OL.39.002012 [22] Craig Gidney. ``Stim: a fast stabilizer circuit simulator''. Quantum 5, 497 (2021). Appearances:. https://doi.org/10.22331/q-2021-07-06-497 [23] Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. ``Decoding across the quantum low-density parity-check code landscape''. Phys. Rev. Res. 2, 043423 (2020). Appearances:. https://doi.org/10.1103/PhysRevResearch.2.043423 [24] Nikolaos Koukoulekidis, Fedor Šimkovic IV, Martin Leib, and Francisco Revson Fernandes Pereira. ``Small quantum codes from algebraic extensions of generalized bicycle codes'' (2024). arXiv:2401.07583. Appearances:. arXiv:2401.07583 [25] Lukas Voss, Sim Jian Xian, Tobias Haug, and Kishor Bharti. ``Multivariate bicycle codes''. Phys. Rev. A 111, L060401 (2025). Appearances:. https://doi.org/10.1103/ll5p-z88p [26] Mackenzie H. Shaw and Barbara M. Terhal. ``Lowering connectivity requirements for bivariate bicycle codes using morphing circuits''. Phys. Rev. Lett. 134, 090602 (2025). Appearances:. https://doi.org/10.1103/PhysRevLett.134.090602Cited byCould not fetch Crossref cited-by data during last attempt 2026-02-23 13:36:47: Could not fetch cited-by data for 10.22331/q-2026-02-23-2009 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-02-23 13:36:47: 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.