  A simple alternative scheme for implementing quantum gates with a single
trapped cold two-level ion beyond the Lamb-Dicke (LD) limit is proposed. Basing
on the quantum dynamics for the laser-ion interaction described by a
generalized Jaynes-Cummings model, one can introduce two kinds of elementary
quantum operations i.e., the simple rotation on the bare atomic state,
generated by applying a resonant pulse, and the joint operation on the internal
and external degrees of the ion, performed by using an off-resonant pulse.
Several typical quantum gates, including Hadamard gate, controlled-Z and
controlled-NOT gates $etc.$, can thus be implemented exactly by using these
elementary operations. The experimental parameters including the LD parameter
and the durations of the applied laser pulses, for these implementation are
derived analytically and numerically. Neither the LD approximation for the
laser-ion interaction nor the auxiliary atomic level is needed in the present
scheme.
