  We introduce a novel parameterization of complex unitary matrices, which
allows for the efficient photonic implementation of arbitrary linear discrete
unitary operators. The proposed architecture is built on factorizing an $N
\times N$ 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 $N+1$ phase layers. We discuss a
gradient-based algorithm for finding the optimal phase parameters for
implementing a given unitary matrix. By increasing the number of phase layers
beyond the critical value of $N+1$, the optimization consistently converges
faster as the system becomes over-determined. We propose an integrated photonic
circuit realization of this architecture with coupled waveguide arrays and
reconfigurable phase modulators. The proposed architecture can pave the way for
developing novel families of programmable photonic circuits for optical
classical and quantum information processing.
