Universal Reconfigurable Linear Photonic Circuits using Discrete Fractional Fourier Transform

Performing fast matrix-vector multiplication near the speed of light is an important task in cutting edge optical classical and quantum information processing and machine learning technologies. On-chip photonic realization of such a device is built on suitable parametrization of arbitrary discrete unitary operations and their decomposition into simple operations that can be realized with simple integrated photonic components. Here, we introduce a novel architecture for parametrization of complex unitary matrices that allows for efficient photonic implementation of arbitrary linear discrete unitary operators. The proposed architecture is built on factorizing an NxN unitary matrix into interlaced discrete fractional Fourier transforms and N-parameter diagonal phase shifts. We show that such a configuration can represent arbitrary unitary operators with N+1 layers of phase modulation. The universality of this architecture is investigated numerically by considering the norm of representation error versus the number of phase layers which results in an abrupt phase transition at N+1 layers. We propose an integrated photonic circuit realization of this architecture with coupled waveguide arrays and reconfigurable phase modulators.