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Error-corrected gates on an encoded qubit


To reach their full potential, quantum computers need to be resilient to noise and decoherence. In such a fault-tolerant quantum computer, errors must be corrected in real time to prevent them from propagating between components1,2. This requirement is especially pertinent while applying quantum gates, where the interaction between components can cause errors to spread quickly throughout the system. However, the large overhead involved in most fault-tolerant architectures2,3 makes implementing these systems a daunting task, motivating the search for hardware-efficient alternatives4,5. Here, we present a gate enacted by an ancilla transmon on a cavity-encoded logical qubit that is fault-tolerant to ancilla decoherence and compatible with logical error correction. We maintain the purity of the encoded qubit by correcting ancilla-induced errors in real time, yielding a reduction of the logical gate error by a factor of two in the presence of naturally occurring decoherence. We also demonstrate a sixfold suppression of the gate error with increased ancilla relaxation errors and a fourfold suppression with increased ancilla dephasing errors. The results demonstrate that bosonic logical qubits can be controlled by error-prone ancilla qubits without inheriting the ancilla’s inferior performance. As such, error-corrected ancilla-enabled gates are an important step towards fault-tolerant processing of bosonic qubits.

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Fig. 1: Working principle of the error-corrected logical gate.
Fig. 2: Experimental protocol and tomography of logical states after gate application.
Fig. 3: Benchmarking of the logical gate.

Data availability

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


  1. 1.

    Preskill, J. Fault-tolerant quantum computation. In Introduction to Quantum Computation and Information (ed. Lo, H.-K.) 213–269 (World Scientific, 1998).

  2. 2.

    Campbell, E. T., Terhal, B. M. & Vuillot, C. Roads towards fault-tolerant universal quantum computation. Nature 549, 172–179 (2017).

    ADS  Article  Google Scholar 

  3. 3.

    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).

    ADS  Article  Google Scholar 

  4. 4.

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

    ADS  Article  Google Scholar 

  5. 5.

    Guillaud, J. & Mirrahimi, M. Repetition cat qubits for fault-tolerant quantum computation. Phys. Rev. X 91, 041053 (2019).

    Google Scholar 

  6. 6.

    Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015).

    ADS  Article  Google Scholar 

  7. 7.

    Nigg, D. et al. Quantum computations on a topologically encoded qubit. Science 345, 302–305 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Cramer, J. et al. Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Nat. Commun. 7, 11526 (2016).

    ADS  Article  Google Scholar 

  9. 9.

    Córcoles, A. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6, 6979 (2015).

    ADS  Article  Google Scholar 

  10. 10.

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

    ADS  Article  Google Scholar 

  11. 11.

    Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).

    ADS  Article  Google Scholar 

  12. 12.

    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  Google Scholar 

  13. 13.

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

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Linke, N. M. et al. Fault-tolerant quantum error detection. Sci. Adv. 3, e1701074 (2017).

    ADS  Article  Google Scholar 

  15. 15.

    Takita, M., Cross, A. W., Córcoles, A. D., Chow, J. M. & Gambetta, J. M. Experimental demonstration of fault-tolerant state preparation with superconducting qubits. Phys. Rev. Lett. 119, 180501 (2017).

    ADS  Article  Google Scholar 

  16. 16.

    Harper, R. & Flammia, S. T. Fault-tolerant logical gates in the IBM quantum experience. Phys. Rev. Lett. 122, 080504 (2019).

    ADS  Article  Google Scholar 

  17. 17.

    Aharonov, D. & Ben-Or, M. Fault-tolerant quantum computation with constant error rate. SIAM J. Comput. 38, 1207–1282 (2008).

    MathSciNet  Article  Google Scholar 

  18. 18.

    Steane, A. M. & Ibinson, B. Fault-tolerant logical gate networks for Calderbank–Shor–Steane codes. Phys. Rev. A 72, 052335 (2005).

    ADS  Article  Google Scholar 

  19. 19.

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

    ADS  Article  Google Scholar 

  20. 20.

    Eastin, B. & Knill, E. Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett. 102, 110502 (2009).

    ADS  Article  Google Scholar 

  21. 21.

    Webster, P. & Bartlett, S. D. Braiding defects in topological stabiliser codes of any dimension cannot be universal. Preprint at (2018).

  22. 22.

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

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Zhou, X., Leung, D. W. & Chuang, I. L. Methodology for quantum logic gate construction. Phys. Rev. A 62, 052316 (2000).

    ADS  Article  Google Scholar 

  24. 24.

    Ma, W.-L. et al. Path independent quantum gates with noisy ancilla. Preprint at (2019).

  25. 25.

    Kapit, E. Error-transparent quantum gates for small logical qubit architectures. Phys. Rev. Lett. 120, 050503 (2018).

    ADS  Article  Google Scholar 

  26. 26.

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

    Google Scholar 

  27. 27.

    Krastanov, S. et al. Universal control of an oscillator with dispersive coupling to a qubit. Phys. Rev. A 92, 040303 (2015).

    ADS  Article  Google Scholar 

  28. 28.

    Heeres, R. W. et al. Cavity state manipulation using photon-number selective phase gates. Phys. Rev. Lett. 115, 137002 (2015).

    ADS  Article  Google Scholar 

  29. 29.

    Sun, L. et al. Tracking photon jumps with repeated quantum non-demolition parity measurements. Nature 511, 444–448 (2014).

    ADS  Article  Google Scholar 

  30. 30.

    Heeres, R. W. et al. Implementing a universal gate set on a logical qubit encoded in an oscillator. Nat. Commun. 8, 94 (2017).

    ADS  Article  Google Scholar 

  31. 31.

    Magesan, E. et al. Efficient measurement of quantum gate error by interleaved randomized benchmarking. Phys. Rev. Lett. 109, 080505 (2012).

    ADS  Article  Google Scholar 

  32. 32.

    Touzard, S. et al. Gated conditional displacement readout of superconducting qubits. Phys. Rev. Lett. 122, 080502 (2019).

    ADS  Article  Google Scholar 

  33. 33.

    Ma, Y. et al. Error-transparent operations on a logical qubit protected by quantum error correction. Nat. Phys. (2020).

  34. 34.

    Xu, Y. et al. Demonstration of controlled-phase gates between two error-correctable photonic qubits. Phys. Rev. Lett. 124, 120501 (2020).

    ADS  Article  Google Scholar 

  35. 35.

    Gao, Y. Y. et al. Entanglement of bosonic modes through an engineered exchange interaction. Nature 566, 509–512 (2019).

    ADS  Article  Google Scholar 

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We thank N. Frattini and K. Sliwa for providing the Josephson parametric converter and N. Ofek for providing the logic for the field programmable gate array used for the control of this experiment. We thank M. Zhang and Y. Wong for helpful discussions. S.R., L.F. and R.J.S. acknowledge funding support from the US Army Research Office (W911NF-18-1-0212). P.R. and S.R. were supported by the Air Force Office of Scientific Research (FA9550-15-1-0015 and FA9550-14-1-0052).

Author information




P.R. and S.R. fabricated the transmon qubits, assembled the experimental apparatus, performed the experiments and analysed the data under the supervision of L.F. and R.J.S. W.-L.M. and L.J. provided theoretical support. P.R., S.R. and R.J.S. wrote the manuscript with feedback from all authors.

Corresponding authors

Correspondence to Philip Reinhold or Serge Rosenblum.

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Competing interests

L.F. and R.J.S. are co-founders of, and equity shareholders in, Quantum Circuits, Inc. S.R., P.R., L.J., L.F. and R.J.S. are inventors on patent application no. 62/613,974 submitted by Yale University, which covers hardware-efficient fault-tolerant operations with superconducting circuits.

Additional information

Peer review information Nature Physics thanks Tanay Roy and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–5, Table 1 and Discussion.

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Reinhold, P., Rosenblum, S., Ma, WL. et al. Error-corrected gates on an encoded qubit. Nat. Phys. 16, 822–826 (2020).

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