Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms
Department of Computer Science, North Carolina State University, Raleigh, North Carolina, USA
| Published: | 2026-02-23, volume 10, page 2009 |
| Editor: | Alexander Schuckert |
| Eprint: | arXiv:2408.10001v6 |
| Doi: | https://doi.org/10.22331/q-2026-02-23-2009 |
| Citation: | Quantum 10, 2009 (2026). |
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Abstract
Quantum 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.
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[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.134.090602
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[1] Daniel J. Spencer, Andrew Tanggara, Tobias Haug, Derek Khu, and Kishor Bharti, "Qudit low-density parity-check codes", Quantum 10, 2023 (2026).
[2] John Stack, Ming Wang, and Frank Mueller, "Transversal Fault Tolerant Distributed Quantum Computing Operations", arXiv:2504.05611, (2025).
[3] Zijian Liang, Bowen Yang, Joseph T. Iosue, and Yu-An Chen, "Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes", arXiv:2410.11942, (2024).
[4] Zijian Liang, Ke Liu, Hao Song, and Yu-An Chen, "Generalized Toric Codes on Twisted Tori for Quantum Error Correction", PRX Quantum 6 2, 020357 (2025).
[5] Min Ye and Nicolas Delfosse, "Quantum error correction for long chains of trapped ions", Quantum 9, 1920 (2025).
[6] Zijian Liang, Jens Niklas Eberhardt, and Yu-An Chen, "Planar quantum low-density parity-check codes with open boundaries", arXiv:2504.08887, (2025).
[7] Abraham Jacob, Campbell McLauchlan, and Dan E. Browne, "Single-Shot Decoding and Fault-tolerant Gates with Trivariate Tricycle Codes", arXiv:2508.08191, (2025).
[8] Yingli Yang, Guo Zhang, and Ying Li, "Planar fault-tolerant logical measurements with low qubit overhead", arXiv:2506.18061, (2025).
[9] Feroz Ahmed Mian, Owen Gwilliam, and Stefan Krastanov, "Multivariate Multicycle Codes for Complete Single-Shot Decoding", arXiv:2601.18879, (2026).
[10] Guo Zhang, Yuanye Zhu, Xiao Yuan, and Ying Li, "Constant-Overhead Magic State Injection into qLDPC Codes with Error Independence Guarantees", arXiv:2505.06981, (2025).
[11] Ming Wang, Ang Li, and Frank Mueller, "Fully Parallelized BP Decoding for Quantum LDPC Codes Can Outperform BP-OSD", arXiv:2507.00254, (2025).
[12] Zijian Liang and Yu-An Chen, "Self-dual bivariate bicycle codes with transversal Clifford gates", arXiv:2510.05211, (2025).
[13] Benjamin C. B. Symons, Abhishek Rajput, and Dan E. Browne, "Sequences of Bivariate Bicycle Codes from Covering Graphs", arXiv:2511.13560, (2025).
[14] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).
[15] Yitao Feng, Hanyu Xue, Ryohei Kobayashi, Po-Shen Hsin, and Yu-An Chen, "Pauli stabilizer formalism for topological quantum field theories and generalized statistics", arXiv:2601.00064, (2025).
[16] Haipeng Xie, Nobuyuki Yoshioka, Kento Tsubouchi, and Ying Li, "Simple, Efficient, and Generic Post-Selection Decoding for qLDPC codes", arXiv:2601.17757, (2026).
[17] Zijian Liang, Jens Niklas Eberhardt, and Yu-An Chen, "Planar Quantum Low-Density Parity-Check Codes with Open Boundaries", PRX Quantum 6 4, 040330 (2025).
[18] Zijian Liang and Yu-An Chen, "Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes", arXiv:2602.20158, (2026).
[19] Xiang Fang, Jixuan Ruan, Sharanya Prabhu, Ang Li, Travis Humble, Dean Tullsen, and Yufei Ding, "Bridging Superconducting and Neutral-Atom Platforms for Efficient Fault-Tolerant Quantum Architectures", arXiv:2601.10144, (2026).
[20] Angelo Di Bella, "Geometry-induced correlated noise in qLDPC syndrome extraction", arXiv:2604.01040, (2026).
[21] Lily Wang, Andy Zeyi Liu, Ray Li, Aleksander Kubica, and Shouzhen Gu, "Check-weight-constrained quantum codes: Bounds and examples", arXiv:2601.15446, (2026).
[22] Ryan Tiew and Nikolas P. Breuckmann, "Low-Overhead Entangling Gates From Generalised Dehn Twists", IEEE Transactions on Information Theory 71 7, 5452 (2025).
[23] Sahil Khan, Abhinav Anand, Kenneth R. Brown, and Jonathan M. Baker, "Cyclone: Designing Efficient and Highly Parallel QCCD Architectural Codesigns for Fault Tolerant Quantum Memory", arXiv:2511.15910, (2025).
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