  The hybrid approach to quantum computation simultaneously utilizes both
discrete and continuous variables which offers the advantage of higher density
encoding and processing powers for the same physical resources. Trapped ions,
with discrete internal states and motional modes which can be described by
continuous variables in an infinite dimensional Hilbert space, offer a natural
platform for this approach. A nonlinear gate for universal quantum computing
can be implemented with the conditional beam splitter Hamiltonian $|e\rangle
\langle e| ( a^{\dagger} b + a b^{\dagger})$ that swaps the quantum states of
two motional modes, depending on the ion's internal state. We realize such a
gate and demonstrate its applications for quantum state overlap measurements,
single-shot parity measurement, and generation of NOON states.
