Highly degenerate photonic waveguide structures for holonomic computation

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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 $\mathrm{U}(3)$ 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.