Logical multi-qubit entanglement with dual-rail superconducting qubits

Nature Physics (2026)Cite this article Recent advances in quantum error correction in various hardware platforms have demonstrated operation near and beyond the threshold for fault-tolerant quantum computing. However, scaling up to achieve the exponential suppression of logical errors needed for fault tolerance remains challenging. Erasure qubits offer a path towards resource-efficient error correction, which enables the hardware-level detection of dominant error types. Single erasure qubits with dual-rail encoding in superconducting devices have demonstrated high coherence and low single-qubit gate errors with mid-circuit erasure detection. Here we demonstrate the generation of logical multi-qubit entanglement under error-biased protection using pairs of tunable transmons in a superconducting quantum processor. Each dual-rail qubit maintains millisecond-scale coherence times and logical single-qubit gate error rates on the order of 10−5 by using post-selection to mitigate erasure errors. We then demonstrate a logical \(\sqrt{{\rm{iSWAP}}}\) gate and the generation of a logical Bell state by engineering tunable couplings between the logical qubits. Building on this, we synthesize a logical controlled-NOT gate with a process fidelity of 98.1% at a 13% erasure rate, enabling the creation of a three-logical-qubit Greenberger–Horne–Zeilinger state with 93.9% fidelity.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutSource data are provided with this paper. Additional data relevant to this study are available from the corresponding authors upon reasonable requestShor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).Article ADS Google Scholar Calderbank, A. R. & Shor, P. W. Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996).Article ADS Google Scholar Knill, E. & Laflamme, R. Theory of quantum error-correcting codes. Phys. Rev. A 55, 900–911 (1997).Article ADS MathSciNet Google Scholar Terhal, B. M. Quantum error correction for quantum memories. Rev. Mod. Phys. 87, 307–346 (2015).Article ADS MathSciNet Google Scholar Sivak, V. V. et al. Real-time quantum error correction beyond break-even. Nature 616, 50–55 (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 Google Quantum AI. Exponential suppression of bit or phase errors with cyclic error correction. Nature 595, 383–387 (2021).Article Google Scholar Reinhold, P. et al. Error-corrected gates on an encoded qubit. Nat. Phys. 16, 822–826 (2020).Article Google Scholar Lachance-Quirion, D. et al. Autonomous quantum error correction of Gottesman-Kitaev-Preskill states. Phys. Rev. Lett. 132, 150607 (2024).Article ADS Google Scholar Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).Article ADS Google Scholar Google Quantum AI and Collaborators. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2024).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 Postler, L. et al. Demonstration of fault-tolerant universal quantum gate operations. Nature 605, 675–680 (2022).Article ADS Google Scholar Erhard, A. et al. Entangling logical qubits with lattice surgery. Nature 589, 220–224 (2021).Article ADS 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 Graham, T. M. et al. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature 604, 457–462 (2022).Article ADS Google Scholar Evered, S. J. et al. High-fidelity parallel entangling gates on a neutral-atom quantum computer. Nature 622, 268–272 (2023).Article ADS Google Scholar Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2023).Article ADS Google Scholar Xu, Q. et al. Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays. Nat. Phys. 20, 1084–1090 (2024).Article Google Scholar Aliferis, P. & Preskill, J. Fault-tolerant quantum computation against biased noise. Phys. Rev. A 78, 052331 (2008).Article ADS Google Scholar Li, M., Miller, D., Newman, M., Wu, Y. & Brown, K. R. 2D compass codes. Phys. Rev. X 9, 021041 (2019).

Google Scholar Guillaud, J. & Mirrahimi, M. Repetition cat qubits for fault-tolerant quantum computation. Phys. Rev. X 9, 041053 (2019).

Google Scholar Xu, Q. et al. Tailored XZZX codes for biased noise. Phys. Rev. Research 5, 013035 (2023).Article ADS Google Scholar Puri, S. et al. Bias-preserving gates with stabilized cat qubits. Sci. Adv. 6, eaay5901 (2020).Mirrahimi, M. et al. Dynamically protected cat-qubits: a new paradigm for universal quantum computation. New J. Phys. 16, 045014 (2014).Article ADS Google Scholar Grassl, M., Beth, T. & Pellizzari, T. Codes for the quantum erasure channel. Phys. Rev. A 56, 33–38 (1997).Article ADS MathSciNet Google Scholar Bennett, C. H., DiVincenzo, D. P. & Smolin, J. A. Capacities of quantum erasure channels. Phys. Rev. Lett. 78, 3217–3220 (1997).Article ADS MathSciNet Google Scholar Wu, Y., Kolkowitz, S., Puri, S. & Thompson, J. D. Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays. Nat. Commun. 13, 4657 (2022).Kubica, A. et al. Erasure qubits: overcoming the T1 limit in superconducting circuits. Phys. Rev. X 13, 041022 (2023).

Google Scholar Sahay, K., Jin, J., Claes, J., Thompson, J. D. & Puri, S. High-threshold codes for neutral-atom qubits with biased erasure errors. Phys. Rev. X 13, 041013 (2023).

Google Scholar Ma, S. et al. High-fidelity gates and mid-circuit erasure conversion in an atomic qubit. Nature 622, 279–284 (2023).Article ADS Google Scholar Kang, M., Campbell, W. C. & Brown, K. R. Quantum error correction with metastable states of trapped ions using erasure conversion. PRX Quantum 4, 020358 (2023).Article ADS Google Scholar Shim, Y. P. & Tahan, C. Semiconductor-inspired design principles for superconducting quantum computing. Nat. Commun. 7, 11059 (2016).Campbell, D. L. et al. Universal nonadiabatic control of small-gap superconducting qubits. Phys. Rev. X 10, 041051 (2020).

Google Scholar Levine, H. et al. Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons. Phys. Rev. X 14, 011051 (2024).

Google Scholar Teoh, J. D. et al. Dual-rail encoding with superconducting cavities. Proc. Natl Acad. Sci. USA 120, e2221736120 (2023).Koottandavida, A. et al. Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity. Phys. Rev. Lett. 132, 180601 (2024).Article ADS Google Scholar Chou, K. S. et al. A superconducting dual-rail cavity qubit with erasure-detected logical measurements. Nat. Phys. 20, 1454–1460 (2024).Article Google Scholar de Graaf, S. J. et al. A mid-circuit erasure check on a dual-rail cavity qubit using the joint-photon number-splitting regime of circuit QED. npj Quantum Inf. 11, 1 (2025).Mohseni, M. & Lidar, D. A. Direct characterization of quantum dynamics: general theory. Phys. Rev. A 75, 062331 (2007).Article ADS MathSciNet Google Scholar Neeley, M. et al. Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state. Nat. Phys. 4, 523–526 (2008).Article Google Scholar Satzinger, K. J. et al. Quantum control of surface acoustic-wave phonons. Nature 563, 661–665 (2018).Article ADS Google Scholar Satzinger, K. J. et al. Simple non-galvanic flip-chip integration method for hybrid quantum systems. Appl. Phys. Lett. 114, 173501 (2019).Qiu, J. et al. Deterministic quantum state and gate teleportation between distant superconducting chips. Sci. Bull. 70, 351–358 (2025).Article Google Scholar Yang, X. et al. Coupler-assisted leakage reduction for scalable quantum error correction with superconducting qubits. Phys. Rev. Lett. 133, 170601 (2024).Article ADS Google Scholar Niu, J. et al. Low-loss interconnects for modular superconducting quantum processors. Nat. Electron. 6, 235–241 (2023).Article Google Scholar Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).Article ADS Google Scholar Yan, F. et al. Tunable coupling scheme for implementing high-fidelity two-qubit gates. Phys. Rev. Appl. 10, 054062 (2018).Article ADS Google Scholar Xu, Y. et al. High-fidelity, high-scalability two-qubit gate scheme for superconducting qubits. Phys. Rev. Lett. 125, 240503 (2020).Article ADS Google Scholar Sung, Y. et al. Realization of high-fidelity CZ and ZZ-free iSWAP gates with a tunable coupler. Phys. Rev. X 11, 021058 (2021).

Google Scholar Magesan, E., Gambetta, J. M. & Emerson, J. Scalable and robust randomized benchmarking of quantum processes. Phys. Rev. Lett. 106, 180504 (2011).Article ADS Google Scholar Magesan, E. et al. Efficient measurement of quantum gate error by interleaved randomized benchmarking. Phys. Rev. Lett. 109, 080505 (2012).Article ADS Google Scholar Yan, F. et al. Rotating-frame relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven evolution. Nat. Commun. 4, 2337 (2013).Guo, Q. et al. Dephasing-insensitive quantum information storage and processing with superconducting qubits. Phys. Rev. Lett. 121, 130501 (2018).Article ADS Google Scholar Steffen, M. et al. State tomography of capacitively shunted phase qubits with high fidelity. Phys. Rev. Lett. 97, 050502 (2006).Article ADS Google Scholar Sete, E. A., Tripathi, V., Valery, J. A., Lidar, D. & Mutus, J. Y. Error budget of a parametric resonance entangling gate with a tunable coupler. Phys. Rev. Appl. 22, 014059 (2024).Article ADS Google Scholar Wehner, S. et al. Quantum internet: a vision for the road ahead. Science 362, eaam9288 (2018).Zhong, Y. et al. Deterministic multi-qubit entanglement in a quantum network. Nature 590, 571–575 (2021).Article ADS Google Scholar Tóth, G. & Apellaniz, I. Quantum metrology from a quantum information science perspective. J. Phys. A: Math. Theor. 47, 424006 (2014).Article ADS MathSciNet Google Scholar Degen, C., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).Article ADS MathSciNet Google Scholar Niroula, P. et al. Quantum sensing with erasure qubits. Phys. Rev. Lett. 133, 080801 (2024).Article ADS Google Scholar Li, S. et al. Ultrahigh-precision Hamiltonian parameter estimation in a superconducting circuit. Phys. Rev. Lett. 132, 250204 (2024).Article ADS Google Scholar Mehta, N. et al. Bias-preserving and error-detectable entangling operations in a superconducting dual-rail system. Preprint at http://arxiv.org/abs/2503.10935 (2025).Download referencesThis work was supported by the Science, Technology and Innovation Commission of Shenzhen Municipality (grant number KQTD20210811090049034) (Y. Zhong), the National Natural Science Foundation of China (grant numbers 12174178 and 12404582) (Y. Zhong and X.L.), Shenzhen Science and Technology Program (grant number RCBS20231211090815032) (X.L.), Guangdong Provincial Project (grant number 2024QN11X158) (X.L.) and the Innovation Program for Quantum Science and Technology (grant number 2021ZD0301703) (Y. Zhong and S.L.).These authors contributed equally: Wenhui Huang, Xuandong Sun, Jiawei Zhang.International Quantum Academy, Shenzhen, ChinaWenhui Huang, Xuandong Sun, Jiawei Zhang, Zechen Guo, Peisheng Huang, Yongqi Liang, Yiting Liu, Daxiong Sun, Zilin Wang, Yuzhe Xiong, Xiaohan Yang, Jiajian Zhang, Libo Zhang, Ji Chu, Weijie Guo, Ji Jiang, Song Liu, Jingjing Niu, Jiawei Qiu, Ziyu Tao, Yuxuan Zhou, Xiayu Linpeng, Youpeng Zhong & Dapeng YuShenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, ChinaWenhui Huang, Xuandong Sun, Jiawei Zhang, Zechen Guo, Yongqi Liang, Yiting Liu, Daxiong Sun, Yuzhe Xiong, Xiaohan Yang, Jiajian Zhang, Libo Zhang, Ji Jiang & Song LiuGuangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen, ChinaWenhui Huang, Xuandong Sun, Jiawei Zhang, Zechen Guo, Yongqi Liang, Yiting Liu, Daxiong Sun, Yuzhe Xiong, Xiaohan Yang, Jiajian Zhang, Libo Zhang, Ji Jiang & Song LiuSchool of Physics, Ningxia University, Yinchuan, ChinaPeisheng Huang & Zilin WangShenzhen Branch, Hefei National Laboratory, Shenzhen, ChinaSong Liu, Jingjing Niu, Youpeng Zhong & Dapeng YuSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarY. Zhong conceived the experiment and supervised the project. X.L. performed the measurements and analysed the data. W.H. designed and tested the device. X.S. developed the field-programmable gate array program for the custom electronics built by Jiawei Zhang. All authors contributed to the experimental setup, discussions of the results and writing of the manuscript.Correspondence to Xiayu Linpeng or Youpeng Zhong.The authors declare no competing interests.Nature Physics thanks Ondřej Černotík, Benjamin Huard and Eli Levenson-Falk for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.a, Calibration of accumulated single qubit phase during \(\sqrt{{\rm{iSWAP}}}\) gate. The top panel is the pulse sequence and the bottom panel is the measured data. The measured data are the population of \({\left|00\right\rangle }_{{\rm{L}}}\) (\({\left|01\right\rangle }_{{\rm{L}}}\)) as a function of the rotation angle of the second π/2 pulse with the logical qubit first initialized to \({\left|00\right\rangle }_{{\rm{L}}}\) (\({\left|01\right\rangle }_{{\rm{L}}}\)). Solid lines are sinusoid fit. The arrows mark the determined calibrated phases. b, Calibration of the relative phase δ during \(\sqrt{{\rm{iSWAP}}}\) gate. The top panel is the pulse sequence and the bottom panel is the measured data. The measured data are the off-diagonal term of the density matrix, \({\rho }_{{\left|01\right\rangle }_{{\rm{L}}}{\left\langle 10\right|}_{{\rm{L}}}}\), as a function of the relative phase δ used in the four calibration pulses Rz(θi). Solid lines are sinusoid fit. The arrow marks the determined calibrated phase.Source dataThe imaginary part of the process matrix χ for the logical CNOT gate.Source dataSupplementary Sections 1–8, Figs. 1–13 and Tables 1–6.Statistical source data for Fig. 2.Statistical source data for Fig. 3.Statistical source data for Fig. 4.Statistical source data for Extended Data Fig. 1.Statistical source data for Extended Data Fig. 2.Springer 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 permissionsHuang, W., Sun, X., Zhang, J. et al. Logical multi-qubit entanglement with dual-rail superconducting qubits. Nat. Phys. (2026). https://doi.org/10.1038/s41567-026-03211-9Download citationReceived: 08 May 2025Accepted: 04 February 2026Published: 06 March 2026Version of record: 06 March 2026DOI: https://doi.org/10.1038/s41567-026-03211-9Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative