  Programmable linear optical processors (LOPs) can have widespread
applications in computing and information processing due to their capabilities
to implement reconfigurable on-chip linear transformations. A conventional LOP
that uses a mesh of Mach-Zehnder interferometers (MZIs) requires $2N+3$ stages
of phase shifters for $N \times N$ matrices. However, it is beneficial to
reduce the number of phase shifter stages to realize a more compact and
lower-loss LOP, especially when long and lossy electro-optic phase shifters are
used. In this work, we propose a novel structure for LOPs that can implement
arbitrary matrices as long as they can be realized by previous MZI-based
schemes. Through numerical analysis, we further show that the number of phase
shifter stages in the proposed structure can be reduced to $N+2$ and $N+3$ for
a large number of random dense matrices and sparse matrices, respectively. This
work contributes to the realization of compact, low-loss, and energy-efficient
programmable LOPs.
