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Quantum error correction of a qubit encoded in grid states of an oscillator


The accuracy of logical operations on quantum bits (qubits) must be improved for quantum computers to outperform classical ones in useful tasks. One method to achieve this is quantum error correction (QEC), which prevents noise in the underlying system from causing logical errors. This approach derives from the reasonable assumption that noise is local, that is, it does not act in a coordinated way on different parts of the physical system. Therefore, if a logical qubit is encoded non-locally, we can—for a limited time—detect and correct noise-induced evolution before it corrupts the encoded information1. In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a non-local qubit: a superposition of position eigenstates that forms grid states of a single oscillator2. However, the implementation of measurements that reveal this noise-induced evolution of the oscillator while preserving the encoded information3,4,5,6,7 has proved to be experimentally challenging, and the only realization reported so far relied on post-selection8,9, which is incompatible with QEC. Here we experimentally prepare square and hexagonal GKP code states through a feedback protocol that incorporates non-destructive measurements that are implemented with a superconducting microwave cavity having the role of the oscillator. We demonstrate QEC of an encoded qubit with suppression of all logical errors, in quantitative agreement with a theoretical estimate based on the measured imperfections of the experiment. Our protocol is applicable to other continuous-variable systems and, in contrast to previous implementations of QEC10,11,12,13,14, can mitigate all logical errors generated by a wide variety of noise processes and facilitate fault-tolerant quantum computation.

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Fig. 1: Quantum error correction protocol.
Fig. 2: Square code in the steady state of the QEC protocol.
Fig. 3: Initialization and coherence characterization of the logical qubit in the square encoding.
Fig. 4: Convergence to the code manifold, state preparation and coherence in the hexagonal code.

Data availability

The experimental data and numerical simulations presented here are available from the corresponding authors upon request.


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We thank C. Flühmann, J. Home, S. Girvin, L. Jiang and K. Noh for discussions and M. Rooks for fabrication assistance. M.M. thanks the Yale Quantum Institute for hosting him while he was collaborating on this project. The use of facilities was supported by YINQE and the Yale SEAS cleanroom. This research was supported by ARO under grant number W911NF-18-1-0212 and ARO grant number W911NF-16-1-0349.

Author information




P.C.-I., A.E. and S.T. designed and performed the experiment and analysed the data. E.Z.-G. N.E.F., V.V.S., P.R., S.S., R.J.S. and L.F. contributed to the experimental apparatus, and S.P. and M.M. provided theoretical support. M.H.D. supervised the project. P.C.-I., A.E., S.T. and M.H.D. wrote the manuscript. All authors provided suggestions for the experiment, discussed the results and contributed to the manuscript.

Corresponding authors

Correspondence to P. Campagne-Ibarcq, S. Touzard or M. H. Devoret.

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Competing interests

L.F., R.J.S. and M.H.D. are founders of QCI. L.F. and R.J.S. are shareholders of QCI. All authors, except A.E. and E.Z.G., are inventors of patents (USA, Japan and Singapore) related to the subject.

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

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Supplementary Information

This file contains Supplementary Materials, including Supplementary Figures 1–13, Supplementary Tables 1 and 2 and Supplementary References.

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

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