  Transformations on quantum states form a basic building block of every
quantum information system. From photonic polarization to two-level atoms,
complete sets of quantum gates for a variety of qubit systems are well known.
For multi-level quantum systems beyond qubits, the situation is more
challenging. The orbital angular momentum modes of photons comprise one such
high-dimensional system for which generation and measurement techniques are
well-studied. However, arbitrary transformations for such quantum states are
not known. Here we experimentally demonstrate a four-dimensional generalization
of the Pauli X-gate and all of its integer powers on single photons carrying
orbital angular momentum. Together with the well-known Z-gate, this forms the
first complete set of high-dimensional quantum gates implemented
experimentally. The concept of the X-gate is based on independent access to
quantum states with different parities and can thus be easily generalized to
other photonic degrees-of-freedom, as well as to other quantum systems such as
ions and superconducting circuits.
