  Structured light, light tailored in its internal degrees of freedom, has
become topical in numerous quantum and classical information processing
protocols. In this work, we harness the high dimensional nature of structured
light modulated in the transverse spatial degree of freedom to realise an
adaptable scheme for learning unitary operations. Our approach borrows from
concepts in variational quantum computing, where a search or optimisation
problem is mapped onto the task of finding a minimum ground state energy for a
given energy/goal function. We achieve this by a pseudo-random walk procedure
over the parameter space of the unitary operation, implemented with optical
matrix-vector multiplication enacted on arrays of Gaussian modes by exploiting
the partial Fourier transforming capabilities of a cylindrical lens in the
transverse degree of freedom for the measurement. We outline the concept
theoretically, and experimentally demonstrate that we are able to learn optical
unitary matrices for dimensions d = 2, 4, 8 and 16 with average fidelities of
>90%. Our work advances high dimensional information processing and can be
adapted to both process and quantum state tomography of unknown states and
channels.
