  Full quantum tomography of high-dimensional quantum systems is experimentally
infeasible due to the exponential scaling of the number of required
measurements on the number of qubits in the system. However, several ideas were
proposed recently for predicting the limited number of features for these
states, or estimating the expectation values of operators, without the need for
full state reconstruction. These ideas go under the general name of shadow
tomography. Here we provide an experimental demonstration of property
estimation based on classical shadows proposed in [H.-Y. Huang, R. Kueng, J.
Preskill. Nat. Phys. https://doi.org/10.1038/s41567-020-0932-7 (2020)] and
study its performance in the quantum optical experiment with high-dimensional
spatial states of photons. We show on experimental data how this procedure
outperforms conventional state reconstruction in fidelity estimation from a
limited number of measurements.
