  We propose a novel, physically-constrained and differentiable approach for
the generation of D-dimensional qudit states via spontaneous parametric
down-conversion (SPDC) in quantum optics. We circumvent any limitations imposed
by the inherently stochastic nature of the physical process and incorporate a
set of stochastic dynamical equations governing its evolution under the SPDC
Hamiltonian. We demonstrate the effectiveness of our model through the design
of structured nonlinear photonic crystals (NLPCs) and shaped pump beams; and
show, theoretically and experimentally, how to generate maximally entangled
states in the spatial degree of freedom. The learning of NLPC structures offers
a promising new avenue for shaping and controlling arbitrary quantum states and
enables all-optical coherent control of the generated states. We believe that
this approach can readily be extended from bulky crystals to thin Metasurfaces
and potentially applied to other quantum systems sharing a similar Hamiltonian
structures, such as superfluids and superconductors.
