  Optical computing harnesses the speed of light to perform vector-matrix
operations efficiently. It leverages interference, a cornerstone of quantum
computing algorithms, to enable parallel computations. In this work, we
interweave quantum computing with classical structured light by formulating the
process of photonic matrix multiplication using quantum mechanical principles
such as state superposition and subsequently demonstrate a well known
algorithm, namely the Deutsch-Jozsa's algorithm. This is accomplished by
elucidating the inherent tensor product structure within the Cartesian
transverse degrees of freedom of light, which is the main resource for optical
vector-matrix multiplication. To this end, we establish a discrete basis using
localized Gaussian modes arranged in a lattice formation and demonstrate the
operation of a Hadamard Gate. Leveraging the reprogrammable and digital
capabilities of spatial light modulators, coupled with Fourier transforms by
lenses, our approach proves adaptable to various algorithms. Therefore our work
advances the use of structured light for quantum information processing.
