Abstract
Quantum 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.
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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).
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Acknowledgements
We 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.
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K.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.
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Extended data
Extended 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.
Extended 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.
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Wang, K., Lu, Z., Zhang, C. et al. Demonstration of low-overhead quantum error correction codes. Nat. Phys. 22, 308–314 (2026). https://doi.org/10.1038/s41567-025-03157-4
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DOI: https://doi.org/10.1038/s41567-025-03157-4
