  We propose an ion trap implementation of Grover's quantum search algorithm
for an unstructured database of arbitrary length N. The experimental
implementation is appealingly simple because the linear ion trap allows for a
straightforward construction, in a single interaction step and without a
multitude of Hadamard transforms, of the reflection operator, which is the
engine of the Grover algorithm. Consequently, a dramatic reduction in the
number of the required physical steps takes place, to just O(\sqrt{N}), the
same as the number of the mathematical steps. The proposed setup allows for
demonstration of both the original (probabilistic) Grover search and its
deterministic variation, and is remarkably robust to imperfections in the
register initialization.
