  High-dimensional quantum systems offer many advantages over low-dimensional
quantum systems. Meanwhile, unitary transformations on quantum states are
important parts in various quantum information tasks, whereas they become
technically infeasible as the dimensionality increases. The photonic orbital
angular momentum (OAM), which is inherit in the transverse spatial mode of
photons, offers a natural carrier to encode information in high-dimensional
spaces. However, it's even more challenging to realize arbitrary unitary
transformations on the photonic OAM states. Here, by combining the path and OAM
degrees of freedom of a single photon, an efficient scheme to realize arbitrary
unitary transformations on the path-OAM coupled quantum states is proposed. The
proposal reduces the number of required interferometers by approximately one
quarter compared with previous works, while maintaining the symmetric
structure. It is shown that by using OAM-to-path interfaces, this scheme can be
utilized to realize arbitrary unitary transformations on the OAM states of
photons. This work facilitates the development of high-dimension quantum state
transformations, and opens a new door to the manipulation of the photonic OAM
states.
