Demonstration of low-overhead quantum error correction codes - Nature

AbstractQuantum computers hold the potential to surpass classical computers in solving complex computational problems. The fragility of quantum information and the error-prone nature of quantum operations necessitate the use of quantum error correction codes to achieve fault-tolerant quantum computing. However, most codes that have been demonstrated so far suffer from low encoding efficiency, and their scalability is hindered by prohibitively high resource overheads. Here we use a 32-qubit quantum processor to demonstrate two low-overhead quantum low-density parity-check codes, a distance-4 bivariate bicycle code and a distance-3 punctured bivariate bicycle code. Utilizing a two-dimensional architecture with overlapping long-range couplers connecting the qubits, we demonstrate the simultaneous measurements of all non-local weight-6 stabilizers via the periodic execution of an efficient syndrome extraction circuit. We achieve a logical error rate per logical qubit per cycle of (8.91 ± 0.17)% for the bivariate bicycle code with four logical qubits and (7.77 ± 0.12)% for the punctured bivariate bicycle code with six logical qubits. Our results establish the feasibility of performing quantum error correction with long-range coupled superconducting processors, demonstrating the viability of low-overhead quantum error correction. Access through your institution Buy or subscribe This is a preview of subscription content, access via your institution Access options Access through your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription 27,99 € / 30 days cancel any time Learn more Subscribe to this journal Receive 12 print issues and online access 251,40 € per year only 20,95 € per issue Learn more Buy this articlePurchase on SpringerLinkInstant access to the full article PDF.39,95 €Prices may be subject to local taxes which are calculated during checkout Fig. 1: Implementation of the BB code.Fig. 2: Error detection in the BB code.Fig. 3: Logical error rate per cycle and performance prediction. Similar content being viewed by others Realizing repeated quantum error correction in a distance-three surface code Article 25 May 2022 Suppressing quantum errors by scaling a surface code logical qubit Article Open access 22 February 2023 High-threshold and low-overhead fault-tolerant quantum memory Article Open access 27 March 2024 Data availability The data presented in the figures and that support the other findings of this study are publicly available via Zenodo at https://doi.org/10.5281/zenodo.17706106 (ref. 53). Source data are provided with this paper. Code availability The data analysis and numerical simulation codes for this study are publicly available via Zenodo at https://doi.org/10.5281/zenodo.17706106 (ref. 53). ReferencesNielsen, M. A., Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).

Google Scholar Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).Article ADS Google Scholar Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).Article ADS MathSciNet Google Scholar Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).Article ADS Google Scholar Herman, D. et al. Quantum computing for finance. Nat. Rev. Phys. 5, 450–465 (2023).Article Google Scholar Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).Article ADS Google Scholar Das Sarma, S., Deng, D.-L. & Duan, L.-M. Machine learning meets quantum physics. Phys. Today 72, 48–54 (2019).Article Google Scholar Gottesman, D. E. Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology (1997).Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002).Article ADS MathSciNet Google Scholar Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).Article ADS Google Scholar Egan, L. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281–286 (2021).Article ADS Google Scholar Ryan-Anderson, C. et al. Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X 11, 041058 (2021). Google Scholar de Neeve, B., Nguyen, T.-L., Behrle, T. & Home, J. P. Error correction of a logical grid state qubit by dissipative pumping. Nat. Phys. 18, 296–300 (2022).Article Google Scholar Paetznick, A. et al. Demonstration of logical qubits and repeated error correction with better-than-physical error rates. Preprint at https://arxiv.org/abs/2404.02280 (2024).Waldherr, G. et al. Quantum error correction in a solid-state hybrid spin register. Nature 506, 204–207 (2014).Article ADS Google Scholar Abobeih, M. H. et al. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature 606, 884–889 (2022).Article ADS Google Scholar Chang, X.-Y. et al. Hybrid entanglement and bit-flip error correction in a scalable quantum network node. Nat. Phys. 22, 583–589 (2025).Article Google Scholar Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).Article ADS Google Scholar Reichardt, B. W. et al. Logical computation demonstrated with a neutral atom quantum processor. Preprint at https://arxiv.org/abs/2411.11822 (2024).Marques, J. F. et al. Logical-qubit operations in an error-detecting surface code. Nat. Phys. 18, 80–86 (2022).Article Google Scholar Zhao, Y. et al. Realization of an error-correcting surface code with superconducting qubits. Phys. Rev. Lett. 129, 030501 (2022).Article ADS Google Scholar Krinner, S. et al. Realizing repeated quantum error correction in a distance-three surface code. Nature 605, 669–674 (2022).Article ADS Google Scholar Google Quantum AI et al. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).Article ADS Google Scholar Ni, Z. et al. Beating the break-even point with a discrete-variable-encoded logical qubit. Nature 616, 56–60 (2023).Article ADS Google Scholar Sivak, V. V. et al. Real-time quantum error correction beyond break-even. Nature 616, 50–55 (2023).Article ADS Google Scholar Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024).Article ADS Google Scholar Lacroix, N. et al. Scaling and logic in the colour code on a superconducting quantum processor. Nature 645, 614–619 (2025).Article ADS Google Scholar Caune, L. et al. Demonstrating real-time and low-latency quantum error correction with superconducting qubits. Preprint at https://arxiv.org/abs/2410.05202 (2024).Eickbusch, A. et al. Demonstration of dynamic surface codes. Nat. Phys. 21, 1994–2001 (2025).Article Google Scholar Google Quantum AI and Collaborators et al. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2025).Article ADS Google Scholar Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).Article ADS MathSciNet Google Scholar Bravyi, S. B. & Kitaev, A. Y. Quantum codes on a lattice with boundary. Preprint at https://arxiv.org/abs/quant-ph/9811052 (1998).Breuckmann, N. P. & Eberhardt, J. N. Quantum low-density parity-check codes. PRX Quantum 2, 040101 (2021).Article ADS Google Scholar Bravyi, S. et al. High-threshold and low-overhead fault-tolerant quantum memory. Nature 627, 778–782 (2024).Article ADS Google Scholar Tillich, J.-P. & Zémor, G. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength. IEEE Trans. Inf. Theory 60, 1193–1202 (2014).Article ADS MathSciNet Google Scholar Panteleev, P. & Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In Proc. 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022) 375–388 (ACM, 2022).Leverrier, A. & Zemor, G. Quantum Tanner codes. In Proc. 63rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2022) 872–883 (IEEE, 2022).Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023).Article ADS Google Scholar Jin, F. et al. Topological prethermal strong zero modes on superconducting processors. Nature 645, 626–632 (2025).Article ADS Google Scholar Roffe, J., White, D. R., Burton, S. & Campbell, E. Decoding across the quantum low-density parity-check code landscape. Phys. Rev. Res. 2, 043423 (2020).Article Google Scholar Panteleev, P. & Kalachev, G. Degenerate quantum LDPC codes with good finite length performance. Quantum 5, 585 (2021).Article Google Scholar Chen, Z. et al. Fabrication and characterization of aluminum airbridges for superconducting microwave circuits. Appl. Phys. Lett. 104, 052602 (2014).Article ADS Google Scholar Song, C. et al. Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits. Science 365, 574–577 (2019).Article ADS MathSciNet Google Scholar Google Quantum AI et al. Exponential suppression of bit or phase errors with cyclic error correction. Nature 595, 383–387 (2021).Article Google Scholar Jeffrey, E. et al. Fast accurate state measurement with superconducting qubits. Phys. Rev. Lett. 112, 190504 (2014).Article ADS Google Scholar Heinsoo, J. et al. Rapid high-fidelity multiplexed readout of superconducting qubits. Phys. Rev. Appl. 10, 034040 (2018).Article ADS Google Scholar Sunada, Y. et al. Fast readout and reset of a superconducting qubit coupled to a resonator with an intrinsic Purcell filter. Phys. Rev. Appl. 17, 044016 (2022).Article ADS Google Scholar Spring, P. A. et al. Fast multiplexed superconducting qubit readout with intrinsic Purcell filtering using a multiconductor transmission line. PRX Quantum 6, 020345 (2025).Article ADS Google Scholar Sank, D. et al. System characterization of dispersive readout in superconducting qubits. Phys. Rev. Appl. 23, 024055 (2025).Article ADS Google Scholar McEwen, M. et al. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat. Commun. 12, 1761 (2021).Article ADS Google Scholar Miao, K. C. et al. Overcoming leakage in quantum error correction. Nat. Phys. 19, 1780–1786 (2023).Article Google Scholar Bluvstein, D. et al. A fault-tolerant neutral-atom architecture for universal quantum computation. Nature 649, 39–46 (2026).Article Google Scholar Wang, K. et al. Demonstration of low-overhead quantum error correction codes. Zenodo https://doi.org/10.5281/zenodo.17706106 (2025).Download referencesAcknowledgementsWe thank D. Yuan and W. Jiang for helpful discussions. The device was fabricated at the Micro-Nano Fabrication Center of Zhejiang University. We acknowledge support from the Quantum Science and Technology–National Science and Technology Major Project (grant numbers 2021ZD0300200 and 2021ZD0302203), the National Natural Science Foundation of China (grant numbers 92365301, 12174342, 12274367, 12322414, 12274368, 12075128, 12404570, 12404574, T2225008 and T24B2002), the Shanghai Qi Zhi Institute Innovation Program SQZ202318, the National Key R&D Program of China (grant number 2023YFB4502600) and the Zhejiang Provincial Natural Science Foundation of China (grant numbers LDQ23A040001 and LR24A040002). In addition, Z.-Z.S., W.L. and D.-L.D. are supported by Tsinghua University Dushi Program. P.-X.S. acknowledges support from the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie grant agreement number 101180589 (SymPhysAI), the National Science Centre (Poland) OPUS grant number 2021/41/B/ST3/04475 and the ‘MagTop’ project (FENG.02.01-IP.05-0028/23) carried out within the ‘International Research Agendas’ programme of the Foundation for Polish Science co-financed by the European Union under the European Funds for Smart Economy 2021-2027 (FENG). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.Author informationAuthor notesThese authors contributed equally: Ke Wang, Zhide Lu, Chuanyu Zhang.Authors and AffiliationsSchool of Physics, ZJU-Hangzhou Global Scientific and Technological Innovation Center, and Zhejiang Key Laboratory of Micro-nano Quantum Chips and Quantum Control, Zhejiang University, Hangzhou, ChinaKe Wang, Chuanyu Zhang, Gongyu Liu, Jiachen Chen, Yanzhe Wang, Yaozu Wu, Shibo Xu, Xuhao Zhu, Feitong Jin, Yu Gao, Ziqi Tan, Zhengyi Cui, Ning Wang, Yiren Zou, Aosai Zhang, Tingting Li, Fanhao Shen, Jiarun Zhong, Zehang Bao, Zitian Zhu, Yihang Han, Yiyang He, Jiayuan Shen, Han Wang, Jia-Nan Yang, Zixuan Song, Jinfeng Deng, Hang Dong, Pengfei Zhang, Hekang Li, Qiujiang Guo, Zhen Wang, Chao Song & H. WangShanghai Qi Zhi Institute, Shanghai, ChinaZhide Lu & Dong-Ling DengCenter for Quantum Information, IIIS, Tsinghua University, Beijing, ChinaZhide Lu, Zheng-Zhi Sun, Weikang Li, Qi Ye, Si Jiang, Yixuan Ma & Dong-Ling DengSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USAQi YeSchool of Physics, Xi’an Jiaotong University, Xi’an, ChinaYixuan MaInternational Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Warsaw, PolandPei-Xin ShenHefei National Laboratory, Hefei, ChinaQiujiang Guo, Zhen Wang, Chao Song, H. Wang & Dong-Ling DengAuthorsKe WangView author publicationsSearch author on:PubMed Google ScholarZhide LuView author publicationsSearch author on:PubMed Google ScholarChuanyu ZhangView author publicationsSearch author on:PubMed Google ScholarGongyu LiuView author publicationsSearch author on:PubMed Google ScholarJiachen ChenView author publicationsSearch author on:PubMed Google ScholarYanzhe WangView author publicationsSearch author on:PubMed Google ScholarYaozu WuView author publicationsSearch author on:PubMed Google ScholarShibo XuView author publicationsSearch author on:PubMed Google ScholarXuhao ZhuView author publicationsSearch author on:PubMed Google ScholarFeitong JinView author publicationsSearch author on:PubMed Google ScholarYu GaoView author publicationsSearch author on:PubMed Google ScholarZiqi TanView author publicationsSearch author on:PubMed Google ScholarZhengyi CuiView author publicationsSearch author on:PubMed Google ScholarNing WangView author publicationsSearch author on:PubMed Google ScholarYiren ZouView author publicationsSearch author on:PubMed Google ScholarAosai ZhangView author publicationsSearch author on:PubMed Google ScholarTingting LiView author publicationsSearch author on:PubMed Google ScholarFanhao ShenView author publicationsSearch author on:PubMed Google ScholarJiarun ZhongView author publicationsSearch author on:PubMed Google ScholarZehang BaoView author publicationsSearch author on:PubMed Google ScholarZitian ZhuView author publicationsSearch author on:PubMed Google ScholarYihang HanView author publicationsSearch author on:PubMed Google ScholarYiyang HeView author publicationsSearch author on:PubMed Google ScholarJiayuan ShenView author publicationsSearch author on:PubMed Google ScholarHan WangView author publicationsSearch author on:PubMed Google ScholarJia-Nan YangView author publicationsSearch author on:PubMed Google ScholarZixuan SongView author publicationsSearch author on:PubMed Google ScholarJinfeng DengView author publicationsSearch author on:PubMed Google ScholarHang DongView author publicationsSearch author on:PubMed Google ScholarZheng-Zhi SunView author publicationsSearch author on:PubMed Google ScholarWeikang LiView author publicationsSearch author on:PubMed Google ScholarQi YeView author publicationsSearch author on:PubMed Google ScholarSi JiangView author publicationsSearch author on:PubMed Google ScholarYixuan MaView author publicationsSearch author on:PubMed Google ScholarPei-Xin ShenView author publicationsSearch author on:PubMed Google ScholarPengfei ZhangView author publicationsSearch author on:PubMed Google ScholarHekang LiView author publicationsSearch author on:PubMed Google ScholarQiujiang GuoView author publicationsSearch author on:PubMed Google ScholarZhen WangView author publicationsSearch author on:PubMed Google ScholarChao SongView author publicationsSearch author on:PubMed Google ScholarH. WangView author publicationsSearch author on:PubMed Google ScholarDong-Ling DengView author publicationsSearch author on:PubMed Google ScholarContributionsK.W. and C.Z. carried out the experiments under the supervision of C.S. and H. Wang. J.C. and Y. Wang designed the device, and J.C. and H.L. fabricated the device under the supervision of H. Wang. Z.W. designed the control and measurement electronics. Z.L. designed the error correction codes and performed the numerical simulations under the supervision of D.-L.D. Z.L., Z.-Z.S., W.L., Q.Y., S.J., Y.M., P.-X.S. and D.-L.D. conducted the theoretical analysis. All authors contributed to the experimental setup, the discussions of the results and the writing of the manuscript.Corresponding authorsCorrespondence to Zhen Wang, Chao Song or Dong-Ling Deng.Ethics declarations Competing interests The authors declare no competing interests. Peer review Peer review information Nature Physics thanks Jean-Claude Besse, Michael Kerschbaum, Matthew McEwen and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Additional informationPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Extended dataExtended Data Fig. 1 Quantum circuit used to perform repeated stabilizer measurements for the [[18,6,3]] punctured BB code.A full syndrome cycle comprises seven layers of CZ gates, interleaved with single-qubit operations, to achieve the simultaneous extraction of all X- and Z-type stabilizers.Extended Data Fig. 2 Detection probability for stabilizers over seven cycles for the [[18,6,3]] punctured BB code.Each data point is obtained from over 40,000 experimental instances. The dotted lines indicate the detection probability for each individual stabilizer, and the solid line shows the average detection probability across all stabilizers of the Z-type or X-type.Source dataExtended Data Fig. 3 Accumulated logical error probabilities as functions of the number of cycles for the [[18,6,3]] punctured BB code.Each data point represents over 40,000 experimental instances after leakage rejection.Source dataExtended Data Table 1 Parameters of bivariate bicycle codes used for logical performance projectionFull size tableSupplementary informationSupplementary InformationSupplementary Figs. 1–13, Tables 1–4 and Discussion.Supplementary Data 1Source data for Supplementary Figures.Source dataSource Data Fig. 1Statistical source data.Source Data Fig. 2Statistical source data.Source Data Fig. 3Statistical source data.Source Data Extended Data Fig. 2Statistical source data.Source Data Extended Data Fig. 3Statistical source data.Rights and permissionsSpringer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsAbout this articleCite this articleWang, K., Lu, Z., Zhang, C. et al. Demonstration of low-overhead quantum error correction codes. Nat. Phys. (2026). https://doi.org/10.1038/s41567-025-03157-4Download citationReceived: 23 May 2025Accepted: 11 December 2025Published: 22 January 2026Version of record: 22 January 2026DOI: https://doi.org/10.1038/s41567-025-03157-4Share this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.