  High-dimensional encoding of quantum information holds the potential to
greatly increase the computational power of existing devices by enlarging the
accessible state space for fixed register size and by reducing the number of
required entangling gates. However, qudit-based quantum computation remains far
less developed than conventional qubit-based approaches, in particular for
photons, which represent natural multi-level information carriers that play a
crucial role in the development of quantum networks. A major obstacle for
realizing quantum gates between two individual photons is the restriction of
direct interaction between photons in linear media. In particular, essential
logic components for quantum operations such as native qudit-qudit entangling
gates are still missing for optical quantum information processing. Here we
address this challenge by presenting a protocol for realizing an entangling
gate -- the controlled phase-flip (CPF) gate -- for two photonic qudits in
arbitrary dimension. We experimentally demonstrate this protocol by realizing a
four-dimensional qudit-qudit CPF gate, whose decomposition would require at
least 13 two-qubit entangling gates. Our photonic qudits are encoded in orbital
angular momentum (OAM) and we have developed a new active high-precision
phase-locking technology to construct a high-dimensional OAM beam splitter that
increases the stability of the CPF gate, resulting in a process fidelity within
a range of $ [0.64 \pm 0.01, 0.82 \pm 0.01]$. Our experiment represents a
significant advance for high-dimensional optical quantum information processing
and has the potential for wider applications beyond optical system.
