Abstract
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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The experimental data and numerical simulations presented here are available from the corresponding authors upon request.
References
Shor, P. Fault-tolerant quantum computation. In Proc. 37th Conf. Foundations of Computer Science 56–65 (IEEE, 1996).
Gottesman, D., Kitaev, A. & Preskill, J. Encoding a qubit in an oscillator. Phys. Rev. A 64, 012310 (2001).
Travaglione, B. & Milburn, G. J. Preparing encoded states in an oscillator. Phys. Rev. A 66, 052322 (2002).
Pirandola, S., Mancini, S., Vitali, D. & Tombesi, P. Continuous variable encoding by ponderomotive interaction. Eur. Phys. J. D 37, 283–290 (2006).
Vasconcelos, H. M., Sanz, L. & Glancy, S. All-optical generation of states for “encoding a qubit in an oscillator”. Opt. Lett. 35, 3261–3263 (2010).
Terhal, B. & Weigand, D. Encoding a qubit into a cavity mode in circuit QED using phase estimation. Phys. Rev. A 93, 012315 (2016).
Motes, K. R., Baragiola, B. Q., Gilchrist, A. & Menicucci, N. C. Encoding qubits into oscillators with atomic ensembles and squeezed light. Phys. Rev. A 95, 053819 (2017).
Flühmann, C., Negnevitsky, V., Marinelli, M. & Home, J. P. Sequential modular position and momentum measurements of a trapped ion mechanical oscillator. Phys. Rev. X 8, 021001 (2018).
Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).
Waldherr, G. et al. Quantum error correction in a solid-state hybrid spin register. Nature 506, 204–207 (2014).
Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015).
Cramer, J. et al. Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Nat. Commun. 7, 11526 (2016).
Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature 536, 441–445 (2016).
Hu, L. et al. Quantum error correction and universal gate set operation on a binomial bosonic logical qubit. Nat. Phys. 15, 503–508 (2019).
Albert, V. V. et al. Performance and structure of single-mode bosonic codes. Phys. Rev. A 97, 032346 (2018).
Noh, K., Albert, V. V. & Jiang, L. Quantum capacity bounds of gaussian thermal loss channels and achievable rates with Gottesman–Kitaev–Preskill codes. IEEE Trans. Inf. Theory 65, 2563–2582 (2019).
Cahill, K. E. & Glauber, R. J. Ordered expansions in boson amplitude operators. Phys. Rev. 177, 1857–1881 (1969).
Aharonov, Y., Pendleton, H. & Petersen, A. Modular variables in quantum theory. Int. J. Theor. Phys. 2, 213–230 (1969).
Popescu, S. Dynamical quantum non-locality. Nat. Phys. 6, 151–153 (2010).
Reagor, M. et al. Quantum memory with millisecond coherence in circuit QED. Phys. Rev. B 94, 014506 (2016).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. Preprint at https://arxiv.org/abs/quant-ph/9511026 (1995).
Svore, K. M., Hastings, M. B. & Freedman, M. Faster phase estimation. Quant. Inf. Comp. 14, 306–328 (2013).
Weigand, D. J. & Terhal, B. M. Generating grid states from Schrödinger-cat states without postselection. Phys. Rev. A 97, 022341 (2018).
Haroche, S. & Raimond, J.-M. Exploring the Quantum (Oxford Univ. Press, 2006).
Walls, D. F. & Milburn, G. J. Quantum Optics (Springer Science & Business Media, 2007).
Glancy, S. & Knill, E. Error analysis for encoding a qubit in an oscillator. Phys. Rev. A 73, 012325 (2006).
Fukui, K., Tomita, A., Okamoto, A. & Fujii, K. High-threshold fault-tolerant quantum computation with analog quantum error correction. Phys. Rev. X 8, 021054 (2018).
Vuillot, C., Asasi, H., Wang, Y., Pryadko, L. P. & Terhal, B. M. Quantum error correction with the toric Gottesman–Kitaev–Preskill code. Phys. Rev. A 99, 032344 (2019).
Touzard, S. et al. Gated conditional displacement readout of superconducting qubits. Phys. Rev. Lett. 122, 080502 (2019).
Puri, S. et al. Stabilized cat in driven nonlinear cavity: a fault-tolerant error syndrome detector. Phys. Rev. X 9, 041009 (2019).
Grimm, A. et al. The Kerr-cat qubit: stabilization, readout, and gates. Nature 584, 205–209 (2020).
Shi, Y., Chamberland, C. & Cross, A. W. Fault-tolerant preparation of approximate GKP states. Preprint at https://arxiv.org/abs/1905.00903 (2019).
Gao, Y. Y. et al. Programmable interference between two microwave quantum memories. Phys. Rev. X 8, 021073 (2018).
Baragiola, B. Q., Pantaleoni, G., Alexander, R. N., Karanjai, A. & Menicucci, N. C. All-Gaussian universality and fault tolerance with the Gottesman–Kitaev–Preskill code. Preprint at https://arxiv.org/abs/1903.00012 (2019).
Acknowledgements
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.
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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.
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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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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). https://doi.org/10.1038/s41586-020-2603-3
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DOI: https://doi.org/10.1038/s41586-020-2603-3
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