  We provide a detailed theoretical and conceptual study of a planned
experiment to excite Rydberg states of ions trapped in a Paul trap. The
ultimate goal is to exploit the strong state dependent interactions between
Rydberg ions to implement quantum information processing protocols and to
simulate the dynamics of strongly interacting spin systems. We highlight the
promises of this approach when combining the high degree of control and readout
of quantum states in trapped ion crystals with the novel and fast gate schemes
based on interacting giant Rydberg atomic dipole moments. We discuss
anticipated theoretical and experimental challenges on the way towards its
realization.
