  We investigate an all-out optical setup allowing for generation of
non-Abelian geometric phases on its large degenerate eigenspaces. The proposal
has the form of an M -pod system and can be implemented in terms of integrated
photonic waveguide structures. We show that by injecting a larger number of
photons into the optical setup, the degeneracy of eigenspaces scales rapidly.
After studying the spectral properties of our system for the general case, we
show how arbitrary U(4) transformations can be generated on the dark subspace
of an optical tripod filled with two photons. Moreover, a degeneracy in the
bright subspaces of the system, absent in any atomic analogue, allows for the
generation of universal single-qubit manipulations. Finally, we address the
complexity issue of holonomic computation. Particularly, we show how two-qubit
and three-qubit states can be implemented on a photonic tripod, where a natural
multi-partite structure is inherited from the spatial mode structure of the
waveguides.
