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  • Review Article
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Learning quantum systems

Nature Reviews Physics volume 5, pages 141–156 (2023)Cite this article

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Abstract

The future development of quantum technologies relies on creating and manipulating quantum systems of increasing complexity, with key applications in computation, simulation and sensing. This poses severe challenges in the efficient control, calibration and validation of quantum states and their dynamics. Although the full simulation of large-scale quantum systems may only be possible on a quantum computer, classical characterization and optimization methods still play an important role. Here, we review different approaches that use classical post-processing techniques, possibly combined with adaptive optimization, to learn quantum systems, their correlation properties, dynamics and interaction with the environment. We discuss theoretical proposals and successful implementations across different multiple-qubit architectures such as spin qubits, trapped ions, photonic and atomic systems, and superconducting circuits. This Review provides a brief background of key concepts recurring across many of these approaches with special emphasis on the Bayesian formalism and neural networks.

Key points

  • The complexity of quantum systems increases exponentially with their size, but in many practical contexts there are assumptions (such as low rank, sparsity, or a specific type of expected dynamics) that enable classical algorithms to be efficient in the characterization of quantum states, dynamics and measurements.

  • Bayesian inference provides a robust approach to learning models for quantum systems, while preserving physical intuition about the processes involved.

  • Neural networks enable the characterization of complex systems, at the expense of physical intuition.

  • Adaptive approaches, which select the next measurement based on knowledge acquired earlier in the estimation, can speed up the characterization process.

  • Classical optimization and learning algorithms can improve the performance of quantum sensors.

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Fig. 1: Learning quantum states, dynamics and measurements.
Fig. 2: Different examples of neural networks for learning quantum systems.
Fig. 3: Bayesian techniques for quantum model learning.
Fig. 4: Optimizing quantum experiments.

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Acknowledgements

The authors thank C. Ferrie for discussions. C.B. is supported by the Engineering and Physical Sciences Research Council (EPSRC) (EP/S000550/1 and EP/V053779/1), the Leverhulme Trust (RPG-2019-388) and the European Commission (QuanTELCO, grant agreement no. 862721). N.A. acknowledges support by the Royal Society (URF/R1/191150), EPSRC Platform grant (EP/R029229/1), the European Research Council (grant agreement 948932) and FQXi grant no. FQXI-IAF19-01. L.B.’s work is supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Superconducting Quantum Materials and Systems Center (SQMS) under contract no. DE-AC02-07CH11359, and by the INFN via the QubIT, SFT and INFN-ML projects. V.G. and L.P. acknowledge financial support from the European Union’s Horizon 2020 research and innovation programme-Qombs Project, FET Flagship on Quantum Technologies grant no. 820419.

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Authors and Affiliations

  1. Istituto Nazionale di Ottica-Consiglio Nazionale delle Ricerche, INO-CNR, Firenze, Italy

    Valentin Gebhart & Luca Pezzè

  2. European Laboratory for Non-Linear Spectroscopy, LENS, Sesto Fiorentino, Italy

    Valentin Gebhart & Luca Pezzè

  3. Boehringer Ingelheim, Quantum Lab, Vienna, Austria

    Raffaele Santagati

  4. PASQAL SAS, Massy, France

    Antonio Andrea Gentile

  5. School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh, UK

    Erik M. Gauger & Cristian Bonato

  6. Department of Materials, University of Oxford, Oxford, UK

    David Craig

  7. Department of Engineering Science, University of Oxford, Oxford, UK

    Natalia Ares

  8. Department of Physics and Astronomy, University of Florence, Florence, Italy

    Leonardo Banchi

  9. Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Firenze, Florence, Italy

    Leonardo Banchi

  10. Max Planck Institute for the Science of Light, Erlangen, Germany

    Florian Marquardt

  11. Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

    Florian Marquardt

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  1. Valentin Gebhart
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The authors contributed equally to all aspects of the article.

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Gebhart, V., Santagati, R., Gentile, A.A. et al. Learning quantum systems. Nat Rev Phys 5, 141–156 (2023). https://doi.org/10.1038/s42254-022-00552-1

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