  An arbitrary lossless transformation in high-dimensional quantum space can be
decomposed into elementary operations which are easy to implement, and an
effective decomposition algorithm is important for constructing
high-dimensional systems. Here, we present two optimized architectures to
effectively realize an arbitrary unitary transformation by using the photonic
path and polarization based on the existing decomposition algorithm. In the
first architecture, the number of required interferometers is reduced by half
compared with previous works. In the second architecture, by using the
high-dimensional X gate, all the elementary operations are transferred to the
operations which act locally on the photonic polarization in the same path.
Such an architecture could be of significance in polarization-based
applications. Both architectures maintain the symmetric layout. Our work
facilitates the optical implementation of high-dimensional transformations and
could have potential applications in high-dimensional quantum computation and
quantum communication.
