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Predicting many properties of a quantum system from very few measurements

Nature Physics volume 16, pages 1050–1057 (2020)Cite this article

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Abstract

Predicting the properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a ‘classical shadow’, can be used to predict many different properties; order \({\mathrm{log}}\,(M)\) measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

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Fig. 1: An illustration for constructing a classical representation, the classical shadow, of a quantum system from randomized measurements.
Fig. 2: Predicting quantum fidelities using classical shadows (Clifford measurements) and NNQST.
Fig. 3: Predicting two-point correlation functions using classical shadows (Pauli measurements) and NNQST.
Fig. 4: Predicting entanglement Rényi entropies using classical shadows (Pauli measurements) and the Brydges et al. protocol.
Fig. 5: Application of classical shadows (Pauli measurements) to variational quantum simulation of the lattice Schwinger model.

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Data availability

Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Source code for an efficient implementation of the proposed procedure is available at https://github.com/momohuang/predicting-quantum-properties.

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Acknowledgements

We thank V. Albert, F. Brandão, M. Endres, I. Roth, J. Tropp, T. Vidick, M. Weilenmann and J. Wright for valuable input and inspiring discussions. L. Aolita and G. Carleo provided helpful advice regarding presentation. Our gratitude extends, in particular, to J. Iverson, who helped us in devising a numerical sampling strategy for toric code ground states. We also thank M. Paini and A. Kalev for informing us about their related work30, where they discussed succinct classical ‘snapshots’ of quantum states obtained from randomized local measurements. H.-Y.H. is supported by the Kortschak Scholars Program. R.K. acknowledges funding provided by the Office of Naval Research (award no. N00014-17-1-2146) and the Army Research Office (award no. W911NF121054). J.P. acknowledges funding from ARO-LPS, NSF and DOE. The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center.

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Authors and Affiliations

  1. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA

    Hsin-Yuan Huang, Richard Kueng & John Preskill

  2. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

    Hsin-Yuan Huang, Richard Kueng & John Preskill

  3. Institute for Integrated Circuits, Johannes Kepler University Linz, Linz, Austria

    Richard Kueng

  4. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA

    John Preskill

Authors
  1. Hsin-Yuan Huang
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  2. Richard Kueng
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  3. John Preskill
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Contributions

H.-Y.H. and R.K. developed the theoretical aspects of this work. H.-Y.H. conducted the numerical experiments and wrote the open-source code. J.P. conceived the applications of classical shadows. H.-Y.H., R.K. and J.P. wrote the manuscript.

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Correspondence to Hsin-Yuan Huang.

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

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Huang, HY., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020). https://doi.org/10.1038/s41567-020-0932-7

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