  Quantum gates are essential for the realization of quantum computer and have
been implemented in various types of two-level systems. However,
high-dimensional quantum gates are rarely investigated both theoretically and
experimentally even that high-dimensional quantum systems exhibit remarkable
advantages over two-level systems for some quantum information and quantum
computing tasks. Here we experimentally demonstrate the four-dimensional X gate
and its unique higher orders with the average conversion efficiency 93\%. All
these gates are based on orbital-angular-momentum degree of freedom of single
photons. Besides, a set of controlled quantum gates is implemented by use of
polarization degree of freedom. Our work is an important step towards the goal
of achieving arbitrary high-dimensional quantum circuit and paves a way for the
implementation of high-dimensional quantum communication and computation.
