Nonadiabatic geometric phases are only dependent on the evolution path of a quantum system but independent of the evolution details, and therefore quantum computation based on nonadiabatic geometric phases is robust against control errors. To realize nonadiabatic geometric quantum computation, it is necessary to ensure that the quantum system undergoes a cyclic evolution and the dynamical phases are removed from the total phases. To satisfy these conditions, the evolution paths in previous schemes are usually restricted to some special forms, e.g, orange-slice-shaped loops, which make the paths unnecessarily long in general. In this paper, we put forward an approach to the realization of nonadiabatic geometric quantum computation by which a universal set of nonadiabatic geometric gates can be realized with any desired evolution paths. Our approach makes it possible to realize geometric quantum computation with an economical evolution time so the influence of environment noises on the quantum gates can be minimized further.
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