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Real-time quantum error correction beyond break-even

Nature volume 616, pages 50–55 (2023)Cite this article

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Abstract

The ambition of harnessing the quantum for computation is at odds with the fundamental phenomenon of decoherence. The purpose of quantum error correction (QEC) is to counteract the natural tendency of a complex system to decohere. This cooperative process, which requires participation of multiple quantum and classical components, creates a special type of dissipation that removes the entropy caused by the errors faster than the rate at which these errors corrupt the stored quantum information. Previous experimental attempts to engineer such a process1,2,3,4,5,6,7 faced the generation of an excessive number of errors that overwhelmed the error-correcting capability of the process itself. Whether it is practically possible to utilize QEC for extending quantum coherence thus remains an open question. Here we answer it by demonstrating a fully stabilized and error-corrected logical qubit whose quantum coherence is substantially longer than that of all the imperfect quantum components involved in the QEC process, beating the best of them with a coherence gain of G = 2.27 ± 0.07. We achieve this performance by combining innovations in several domains including the fabrication of superconducting quantum circuits and model-free reinforcement learning.

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Fig. 1: Experimental system.
Fig. 2: QEC implementation and optimization.
Fig. 3: System coherence.
Fig. 4: Analysis of error syndromes.

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Data availability

The data that support the findings of this study are available from the corresponding authors upon a request.

Code availability

The open-source implementation of the proximal policy optimization algorithm is available in ref. 29. The custom code used for quantum control optimization, data acquisition, analysis and visualization is available from the corresponding authors upon a request.

References

  1. Hu, L. et al. Quantum error correction and universal gate set operation on a binomial bosonic logical qubit. Nat. Phys. 15, 503–508 (2019).

    Article  CAS  Google Scholar 

  2. Gertler, J. M. et al. Protecting a bosonic qubit with autonomous quantum error correction. Nature 590, 243–248 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  3. Campagne-Ibarcq, P. et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature 584, 368–372 (2020).

    Article  CAS  PubMed  Google Scholar 

  4. Krinner, S. et al. Realizing repeated quantum error correction in a distance-three surface code. Nature 605, 669–674 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Zhao, Y. et al. Realization of an error-correcting surface code with superconducting qubits. Phys. Rev. Lett. 129, 030501 (2022).

    Article  ADS  CAS  PubMed  Google Scholar 

  6. Sundaresan, N. et al. Matching and maximum likelihood decoding of a multi-round subsystem quantum error correction experiment. Preprint at https://arxiv.org/abs/2203.07205 (2022).

  7. Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).

  8. Knill, E. & Laflamme, R. Theory of quantum error-correcting codes. Phys. Rev. A 55, 900–911 (1997).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  9. Gottesman, D., Kitaev, A. & Preskill, J. Encoding a qubit in an oscillator. Phys. Rev. A 64, 012310 (2001).

    Article  ADS  Google Scholar 

  10. Mirrahimi, M. et al. Dynamically protected cat-qubits: a new paradigm for universal quantum computation. N. J. Phys. 16, 045014 (2014).

    Article  MATH  Google Scholar 

  11. Michael, M. H. et al. New class of quantum error-correcting codes for a bosonic mode. Phys. Rev. X 6, 031006 (2016).

    Google Scholar 

  12. Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).

    Article  ADS  CAS  PubMed  Google Scholar 

  13. Steane, A. M. Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  14. Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

    Article  ADS  Google Scholar 

  15. Noh, K. & Chamberland, C. Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code. Phys. Rev. A 101, 012316 (2020).

    Article  ADS  CAS  Google Scholar 

  16. Terhal, B. M., Conrad, J. & Vuillot, C. Towards scalable bosonic quantum error correction. Quantum Sci. Technol. 5, 043001 (2020).

    Article  ADS  Google Scholar 

  17. Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441–445 (2016).

    Article  ADS  CAS  PubMed  Google Scholar 

  18. 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 

  19. Ryan-Anderson, C. et al. Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X 11, 041058 (2021).

    CAS  Google Scholar 

  20. Egan, L. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281–286 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  21. Waldherr, G. et al. Quantum error correction in a solid-state hybrid spin register. Nature 506, 204–207 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Abobeih, M. et al. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature 606, 884–889 (2022).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  23. Xue, X. et al. Quantum logic with spin qubits crossing the surface code threshold. Nature 601, 343–347 (2022).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  24. Place, A. P. et al. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nat. Commun. 12, 1779 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  25. Reagor, M. et al. Quantum memory with millisecond coherence in circuit QED. Phys. Rev. B 94, 014506 (2016).

    Article  ADS  Google Scholar 

  26. Royer, B., Singh, S. & Girvin, S. M. Stabilization of finite-energy Gottesman-Kitaev-Preskill states. Phys. Rev. Lett. 125, 260509 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  27. Sutton, R. S. & Barto, A. G. Reinforcement Learning: an Introduction (MIT Press, 2018).

  28. Schulman, J., Wolski, F., Dhariwal, P., Radford, A. & Klimov, O. Proximal policy optimization algorithms. Preprint at https://arxiv.org/abs/1707.06347 (2017).

  29. Guadarrama, S. et al. TF-Agents: a library for reinforcement learning in TensorFlow. GitHub https://github.com/tensorflow/agents (2018).

  30. Lescanne, R. et al. Exponential suppression of bit-flips in a qubit encoded in an oscillator. Nat. Phys. 16, 509–513 (2020).

    Article  CAS  Google Scholar 

  31. Lloyd, S. & Viola, L. Engineering quantum dynamics. Phys. Rev. A 65, 010101 (2001).

    Article  ADS  MathSciNet  Google Scholar 

  32. Shen, C. et al. Quantum channel construction with circuit quantum electrodynamics. Phys. Rev. B 95, 134501 (2017).

    Article  ADS  Google Scholar 

  33. Chen, Z. et al. Measuring and suppressing quantum state leakage in a superconducting qubit. Phys. Rev. Lett. 116, 020501 (2016).

    Article  ADS  PubMed  Google Scholar 

  34. Eickbusch, A. et al. Fast universal control of an oscillator with weak dispersive coupling to a qubit. Nat. Phys. 18, 1464–1469 (2022).

    Article  CAS  Google Scholar 

  35. Kelly, J. et al. Optimal quantum control using randomized benchmarking. Phys. Rev. Lett. 112, 240504 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  36. Werninghaus, M. et al. Leakage reduction in fast superconducting qubit gates via optimal control. npj Quantum Inf. 7, 14 (2021).

    Article  ADS  Google Scholar 

  37. Sivak, V. V. et al. Model-free quantum control with reinforcement learning. Phys. Rev. X 12, 011059 (2022).

    CAS  Google Scholar 

  38. Terhal, B. M. & Weigand, D. Encoding a qubit into a cavity mode in circuit QED using phase estimation. Phys.l Rev. A 93, 012315 (2016).

    Article  ADS  Google Scholar 

  39. Hastrup, J. & Andersen, U. L. Improved readout of qubit-coupled Gottesman-Kitaev-Preskill states. Quantum Sci. Technol. 6, 035016 (2021).

    Article  ADS  Google Scholar 

  40. Klimov, P. V. et al. Fluctuations of energy-relaxation times in superconducting qubits. Phys. Rev. Lett. 121, 90502 (2018).

    Article  ADS  CAS  Google Scholar 

  41. Nielsen, M. A. A simple formula for the average gate fidelity of a quantum dynamical operation. Phys. Lett. A 303, 249–252 (2002).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  42. Bravyi, S. & Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A 71, 022316 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Grimm, A. et al. Stabilization and operation of a Kerr-cat qubit. Nature 584, 205–209 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  44. Chen, Z. et al. Exponential suppression of bit or phase errors with cyclic error correction. Nature 595, 383–387 (2021).

    Article  Google Scholar 

  45. Andersen, C. K. et al. Repeated quantum error detection in a surface code. Nat. Phys. 16, 875–880 (2020).

    Article  CAS  Google Scholar 

  46. Ma, W.-L. et al. Path-independent quantum gates with noisy ancilla. Phys. Rev. Lett. 125, 110503 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  47. Rosenblum, S. et al. Fault-tolerant detection of a quantum error. Science 361, 266–270 (2018).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  48. Puri, S. et al. Stabilized cat in a driven nonlinear cavity: a fault-tolerant error syndrome detector. Phys. Rev. X 9, 041009 (2019).

    CAS  Google Scholar 

  49. Ni, Z. et al. Beating the break-even point with a discrete variable-encoded logical qubit. Nature https://doi.org/10.1038/s41586-023-05784-4 (2023).

  50. Gross, J. A., Caves, C. M., Milburn, G. J. & Combes, J. Qubit models of weak continuous measurements: Markovian conditional and open-system dynamics. Quantum Sci. Technol. 3, 024005 (2018).

    Article  ADS  Google Scholar 

  51. Pfaff, W. et al. Controlled release of multiphoton quantum states from a microwave cavity memory. Nat. Phys. 13, 882–887 (2017).

    Article  CAS  Google Scholar 

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Acknowledgements

We acknowledge discussions with R. Cortiñas, J. Claes and A. Mi. We thank B. Huard for feedback on the manuscript. This research was supported by the US Army Research Office under grants W911NF-18-1-0212 and W911NF-16-1-0349, and by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the US Government. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. The use of fabrication facilities was supported by the Yale Institute for Nanoscience and Quantum Engineering and the Yale SEAS Cleanroom.

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Author notes

  1. V. V. Sivak

    Present address: Google AI Quantum, Santa Barbara, CA, USA

Authors and Affiliations

  1. Department of Physics, Yale University, New Haven, CT, USA

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf & M. H. Devoret

  2. Department of Applied Physics, Yale University, New Haven, CT, USA

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf & M. H. Devoret

  3. Yale Quantum Institute, Yale University, New Haven, CT, USA

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf & M. H. Devoret

  4. Institut Quantique, Université de Sherbrooke, Sherbrooke, Quebec, Canada

    B. Royer

  5. Département de Physique, Université de Sherbrooke, Sherbrooke, Quebec, Canada

    B. Royer

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Contributions

V.V.S., A.M. and A.E. built the experimental setup. R.J.S. contributed to experimental apparatus. I.T., S.G. and L.F. fabricated the transmon chip. B.R., S.S. and S.M.G. developed the theory. B.R., V.V.S., A.E. and B.L.B. developed dissipative oscillator cooling. A.E., V.V.S. and A.Z.D. developed the state initialization technique. V.V.S. implemented RL, carried out the experiments and analysed data. V.V.S., A.E., B.R. and M.H.D. regularly discussed the project and provided insight. M.H.D. supervised the project. V.V.S. and M.H.D. wrote the manuscript with feedback from all authors.

Corresponding authors

Correspondence to V. V. Sivak or M. H. Devoret.

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Competing interests

R.J.S., L.F. and M.H.D. are founders, and R.J.S. and L.F. are shareholders of Quantum Circuits, Inc.

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Nature thanks Atsushi Noguchi and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Supplementary information

Supplementary Information

This file contains the following four sections and additional references: I. Experimental setup and sample parameters; II. Calibration and characterization experiments; III. Quantum control optimization; and IV. QEC of the grid code.

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Sivak, V.V., Eickbusch, A., Royer, B. et al. Real-time quantum error correction beyond break-even. Nature 616, 50–55 (2023). https://doi.org/10.1038/s41586-023-05782-6

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