  Predicting 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.
