  Characterization of quantum states is a fundamental requirement in quantum
science and technology. As a promising framework, shadow tomography shows
significant efficiency in estimating linear functions, however, for the
challenging nonlinear ones, it requires measurements at an exponential cost.
Here, we implement an advanced shadow protocol, so-called hybrid shadow~(HS)
tomography, to reduce the measurement cost in the estimation of nonlinear
functions in an optical system. We design and realize a deterministic quantum
Fredkin gate with single photon, achieving high process fidelity of
$0.935\pm0.001$. Utilizing this novel Fredkin gate, we demonstrate HS in the
estimations, like the higher-order moments up to 4, and reveal that the sample
complexity of HS is significantly reduced compared with the original shadow
protocol. Furthermore, we utilize these higher-degree functions to implement
virtual distillation, which effectively extracts a high-purity quantum state
from two noisy copies. The virtual distillation is also verified in a
proof-of-principle demonstration of quantum metrology, further enhancing the
accuracy of parameter estimation. Our results suggest that HS is efficient in
state characterization and promising for quantum technologies.
