Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Predicting many properties of a quantum system from very few measurements

## 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.

## Access options

from\$8.99

All prices are NET prices.

## 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.

## References

1. 1.

Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

2. 2.

Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).

3. 3.

Carrasquilla, J., Torlai, G., Melko, R. G. & Aolita, L. Reconstructing quantum states with generative models. Nat. Mach. Intell. 1, 155–161 (2019).

4. 4.

Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).

5. 5.

Aaronson, S. Shadow tomography of quantum states. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018) 325–338 (ACM, 2018)

6. 6.

Aaronson, S. & Rothblum, G. N. Gentle measurement of quantum states and differential privacy. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019) 322–333 (ACM, 2019).

7. 7.

Guta, M., Kahn, J., Kueng, R. J. & Tropp, J. A. Fast state tomography with optimal error bounds. J. Phys. A 53, 204001 (2020).

8. 8.

Gottesman, D. Stabilizer Codes and Quantum Error Correction PhD thesis, Caltech (1997).

9. 9.

Fano, R. M. Transmission of Information: A Statistical Theory of Communications (MIT Press, 1961).

10. 10.

Jerrum, M. R., Valiant, L. G. & Vazirani, V. V. Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43, 169–188 (1986).

11. 11.

Nemirovsky, A. S. & Yudin, D. B. Problem Complexity and Method Efficiency in Optimization (Wiley-Interscience, 1983).

12. 12.

Greenberger, D. M., Horne, M. A. & Zeilinger, A. in Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Fundamental Theories of Physics Vol. 37 (ed. Kafatos, M.) 69–72 (Springer, 1989).

13. 13.

Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory.J. Math. Phys. 43, 4452–4505 (2002).

14. 14.

Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).

15. 15.

Gühne, O. & Tóth, G. Entanglement detection. Phys. Rep. 474, 1–75 (2009).

16. 16.

Weilenmann, M., Dive, B., Trillo, D., Aguilar, E. A. & Navascués, M. Entanglement detection beyond measuring fidelities. Phys. Rev. Lett. 124, 200502 (2020).

17. 17.

Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

18. 18.

Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

19. 19.

Hoeffding, W. in Breakthroughs in Statistics 308–334 (Springer, 1992).

20. 20.

Brydges, T. et al. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).

21. 21.

Renes, J. M., Blume-Kohout, R., Scott, A. J. & Caves, C. M. Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004).

22. 22.

Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

23. 23.

Dasgupta, C. & Ma, S.-k Low-temperature properties of the random Heisenberg antiferromagnetic chain. Phys. Rev. B 22, 1305 (1980).

24. 24.

Ma, S.-k, Dasgupta, C. & Hu, C.-k Random antiferromagnetic chain. Phys. Rev. Lett. 43, 1434 (1979).

25. 25.

Bonet-Monroig, X., Babbush, R. & O’Brien, T. E. Nearly optimal measurement scheduling for partial tomography of quantum states. Preprint at https://arxiv.org/pdf/1908.05628.pdf (2019).

26. 26.

Raghavan, P. Probabilistic construction of deterministic algorithms: approximating packing integer programs.J. Comput. Syst. Sci. 37, 130–143 (1988).

27. 27.

Spencer, J. Ten lectures on the probabilistic method. In CBMS-NSF Regional Conference Series in Applied Mathematics 2nd edn, Vol. 64 (SIAM, 1994).

28. 28.

Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017).

29. 29.

Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017).

30. 30.

Paini, M. & Kalev, A. An approximate description of quantum states. Preprint at https://arxiv.org/pdf/1910.10543.pdf (2019).

## 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.

## Author information

Authors

### 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.

### Corresponding author

Correspondence to Hsin-Yuan Huang.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–4.

## Rights and permissions

Reprints and Permissions

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

• Accepted:

• Published:

• Issue Date:

• ### Power of data in quantum machine learning

• Hsin-Yuan Huang
• , Michael Broughton
• , Masoud Mohseni
• , Ryan Babbush
• , Sergio Boixo
• , Hartmut Neven
•  & Jarrod R. McClean

Nature Communications (2021)

• ### Experimental Estimation of Quantum State Properties from Classical Shadows

• G.I. Struchalin
• , Ya. A. Zagorovskii
• , E.V. Kovlakov
• , S.S. Straupe
•  & S.P. Kulik

PRX Quantum (2021)

• ### Fast Estimation of Sparse Quantum Noise

• Robin Harper
• , Wenjun Yu
•  & Steven T. Flammia

PRX Quantum (2021)

• ### Learning models of quantum systems from experiments

• Antonio A. Gentile
• , Brian Flynn
• , Sebastian Knauer
• , Nathan Wiebe
• , Stefano Paesani
• , John G. Rarity
• , Raffaele Santagati
•  & Anthony Laing

Nature Physics (2021)

• ### Many-Body Chern Number from Statistical Correlations of Randomized Measurements

• Ze-Pei Cian
• , Hossein Dehghani
• , Andreas Elben
• , Benoît Vermersch
• , Guanyu Zhu
• , Maissam Barkeshli
• , Peter Zoller