  Unitary transformations are the fundamental building blocks of gates and
operations in quantum information processing allowing the complete manipulation
of quantum systems in a coherent manner. In the case of photons, optical
elements that can perform unitary transformations are readily available only
for some degrees of freedom, e.g. wave plates for polarisation. However for
high-dimensional states encoded in the transverse spatial modes of light,
performing arbitrary unitary transformations remains a challenging task for
both theoretical proposals and actual implementations. Following the idea of
multi-plane light conversion, we show that it is possible to perform a broad
variety of unitary operations when the number of phase modulation planes is
comparable to the number of modes. More importantly, we experimentally
implement several high-dimensional quantum gates for up to 5-dimensional states
encoded in the full-field mode structure of photons. In particular, we realise
cyclic and quantum Fourier transformations, known as Pauli $\hat{X}$-gates and
Hadamard $\hat{H}$-gates, respectively, with an average visibility of more than
90%. In addition, we demonstrate near-perfect "unitarity" by means of quantum
process tomography unveiling a process purity of 99%. Lastly, we demonstrate
the benefit of the two independent spatial degrees of freedom, i.e. azimuthal
and radial, and implement a two-qubit controlled-NOT quantum operation on a
single photon. Thus, our demonstrations open up new paths to implement
high-dimensional quantum operations, which can be applied to various tasks in
quantum communication, computation and sensing schemes.
