  All-dielectric optical metasurfaces are a workhorse in nano-optics due to
both their ability to manipulate light in different degrees of freedom and
their excellent performance at light frequency conversion. Here, we demonstrate
first-time generation of photon pairs via spontaneous parametric-down
conversion in lithium niobate quantum optical metasurfaces with electric and
magnetic Mie-like resonances at various wavelengths. By engineering the quantum
optical metasurface, we tailor the photon-pair spectrum in a controlled way.
Within a narrow bandwidth around the resonance, the rate of pair production is
enhanced up to two orders of magnitude compared to an unpatterned film of the
same thickness and material. These results enable flat-optics sources of
entangled photons -- a new promising platform for quantum optics experiments.
