  Linear optical operations are fundamental and significant for both quantum
mechanics and classical technologies. We demonstrate a non-cascaded approach to
perform arbitrary unitary and non-unitary linear operations for N-dimensional
phase-coherent spatial modes with meticulously designed phase gratings. As
implemented on spatial light modulators (SLMs), the unitary transformation
matrix has been realized with dimensionalities ranging from 7 to 24 and the
corresponding fidelities are from 95.1% to 82.1%. For the non-unitary
operators, a matrix is presented for the tomography of a 4-level quantum system
with a fidelity of 94.9%. Thus, the linear operator has been successfully
implemented with much higher dimensionality than that in previous reports. It
should be mentioned that our method is not limited to SLMs and can be easily
applied on other devices. Thus we believe that our proposal provides another
option to perform linear operation with a simple, fixed, error-tolerant and
scalable scheme.
