Predicting many properties of a quantum system from very few measurements

• Published in: Nature physics Vol. 16; no. 10; pp. 1050 - 1057
• Main Authors: Huang, Hsin-Yuan, Kueng, Richard, Preskill, John
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 01.10.2020; Nature Publishing Group
• Subjects: 639/705; 639/766/259; 639/766/483; 639/766/483/481; 639/766/483/640; Approximation; Atomic; Circuits; Classical and Continuum Physics; Complex Systems; Condensed Matter Physics; Entropy; Hamiltonian functions; Information theory; Lower bounds; Mathematical and Computational Physics; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy; Properties (attributes); Quantum entanglement; Quantum theory; Shadows; Theoretical; Tomography
• ISSN: 1745-2473; 1745-2481
• DOI: 10.1038/s41567-020-0932-7

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. An efficient method has been proposed through which the properties of a complex, large-scale quantum system can be predicted without fully characterizing the quantum state.