Experimental Estimation of Quantum State Properties from Classical Shadows

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 have been 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 Huang et al. [Nat. Phys. 16, 1050 (2020)] and study its performance in a quantum-optical experiment with high-dimensional spatial states of photons. We show by means of experimental data how this procedure outperforms conventional state reconstruction in fidelity estimation from a limited number of measurements.