Highly degenerate photonic waveguide structures for holonomic computation

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(3) 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 analog, 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 multipartite structure is inherited from the spatial mode structure of the waveguides.