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Quantum spherical codes

Nature Physics (2024)Cite this article

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Abstract

As with classical computers, quantum computers require error-correction schemes to reliably perform useful large-scale calculations. The nature and frequency of errors depends on the quantum computing platform, and although there is a large literature on qubit-based coding, these are often not directly applicable to devices that store information in bosonic systems such as photonic resonators. Here, we introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, and we obtain multimode extensions of the cat codes that can outperform previous constructions but require a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that, at the same time, form averaging sets known as spherical designs. We also recast concatenations of Calderbank–Shor–Steane codes with cat codes as quantum spherical codes, which establishes a method to autonomously protect against dephasing noise.

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Fig. 1: Quantum spherical codewords are quantum superpositions of constellations on a sphere.

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Data availability

Coordinates of the polytopes used for our codes constitute the minimum dataset necessary to verify the results in this article. Unless otherwise noted, coordinates for each code were obtained from the references linked in the corresponding row of our code tables in the Supplementary Information. Otherwise, we provide the coordinates explicitly in the main text or the Supplementary Information. Explicit coordinates for any particular code are available upon request.

Code availability

MATHEMATICA notebooks generated during the current study are available from the corresponding author on request. Further details and references about spherical codes described in this manuscript are available at the Error-correction Zoo website at http://errorcorrectionzoo.org and via Github at http://github.com/errorcorrectionzoo.

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Acknowledgements

We thank F. Arzani, A. Burchards, J. Conrad, A. Denys, P. Faist, M. Gullans, W. He, L. Jiang, G. Kuperberg, G. Lee, A. Leverrier, P. Panteleev, S. Singh and G. Zheng for helpful discussions. V.V.A. is especially grateful to K. Noh for discussion about realizations of these codes. This work is supported in part by NSF grants OMA-2120757 (QLCI), CCF-2110113 (NSF-BSF), CCF-2104489 and CCF-2330909. J.T.I. thanks the Joint Quantum Institute at the University of Maryland for support through a JQI fellowship. Our figures were drawn using MATHEMATICA 13 following the prescription of ref. 79. Contributions to this work by NIST, an agency of the US government, are not subject to US copyright. Any mention of commercial products does not indicate endorsement by NIST. V.V.A. thanks R. Kandratsenia, Tatyana Albert, Thomas Albert and O. Albert for providing daycare support throughout this work.

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  1. Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD, USA

    Shubham P. Jain, Joseph T. Iosue & Victor V. Albert

  2. Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA

    Joseph T. Iosue

  3. Department of ECE and ISR, University of Maryland, College Park, MD, USA

    Alexander Barg

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Correspondence to Victor V. Albert.

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Jain, S.P., Iosue, J.T., Barg, A. et al. Quantum spherical codes. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02496-y

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