Following a key idea of unconventional geometric quantum computation developed earlier [Phys. Rev. Lett. 91, 197902 (2003)], here we propose a more general scheme in such an intriguing way: γ_d=α_g+ηγ _g, where γ_d and γ_g are respectively the dynamic and geometric phases accumulated in the quantum gate operation, with η as a constant and α_g being dependent only on the geometric feature of the operation. More arrestingly, we demonstrate the first experiment to implement a universal set of such

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... kind of generalized unconventional geometric quantum gates with high fidelity in an NMR system.