Local quantum gates represent the basic building blocks of quantum computers. In the photonic domain, these gates act on individual photons and are usually implemented as single optical devices. In cases when the same gate is to be applied simultaneously to multiple photons, the corresponding physical implementation consists of a stack of identical devices. We present a systematic approach that allows one to replace this stack with a single device supplemented by pre- and post-processing stages. As a result, considerable savings in resources are obtained. We exemplify our scheme for the case of the orbital angular momentum (OAM) of single photons and demonstrate the scheme for high-dimensional Pauli gates and their arbitrary powers. As a byproduct, we also present a simple argument that explicitly shows how to construct an arbitrary quantum gate acting on OAM of single photons, where only conventional optical elements are utilized.