  A crucial subroutine for various quantum computing and communication
algorithms is to efficiently extract different classical properties of quantum
states. In a notable recent theoretical work by Huang, Kueng, and Preskill
[Nat. Phys. 16, 1050 (2020)], a thrifty scheme showed how to project the
quantum state into classical shadows and simultaneously predict $M$ different
functions of a state with only $\mathcal{O}(\log_2 M)$ measurements,
independent of the system size and saturating the information-theoretical
limit. Here, we experimentally explore the feasibility of the scheme in the
realistic scenario with a finite number of measurements and noisy operations.
We prepare a four-qubit GHZ state and show how to estimate expectation values
of multiple observables and Hamiltonians. We compare the measurement strategies
with uniform, biased, and derandomized classical shadows to conventional ones
that sequentially measure each state function exploiting either importance
sampling or observable grouping. We next demonstrate the estimation of
nonlinear functions using classical shadows and analyze the entanglement of the
prepared quantum state. Our experiment verifies the efficacy of exploiting
(derandomized) classical shadows and sheds light on efficient quantum computing
with noisy intermediate-scale quantum hardware.
