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 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.
Additional Information
© 2020 Springer Nature Limited. Received 20 October 2019. Accepted 06 May 2020. Published 22 June 2020. 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. 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. Author 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. The authors declare no competing interests.Attached Files
Submitted - 2002.08953.pdf
Supplemental Material - 41567_2020_932_MOESM1_ESM.pdf
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Additional details
- Eprint ID
- 102787
- Resolver ID
- CaltechAUTHORS:20200427-084340790
- Kortschak Scholars Program
- Office of Naval Research (ONR)
- N00014-17-1-2146
- Army Research Office (ARO)
- W911NF121054
- NSF
- Department of Energy (DOE)
- Created
-
2020-04-27Created from EPrint's datestamp field
- Updated
-
2021-11-16Created from EPrint's last_modified field
- Caltech groups
- Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics