How to realize a universal quantum gate with trapped ions

We report the realization of an elementary quantum processor based on a linear crystal of trapped ions. Each ion serves as a quantum bit (qubit) to store the quantum information in long lived electronic states. We present the realization of single-qubit and of universal two-qubit logic gates. The two-qubit operation relies on the coupling of the ions through their collective quantized motion. A detailed description of the setup and the methods is included.