Trapped strings of cold ions provide an ideal system for quantum information processing. The quantum information can be stored in individual ions and these qubits can be individually prepared; the corresponding quantum states can be manipulated and measured with nearly 100% detection efficiency. With a small ion‐trap quantum computer based on up to fourteen trapped Ca+ ions as qubits we have generated genuine quantum states in a preprogrammed way. In particular, using high fidelity global and local quantum gate operations, with GHZ states of up to fourteen ions we have investigated noise and error sources during quantum information processing. Decoherence of multi‐qubit GHZ states was measured and is quantitatively described a noise model. Using optimized sequences of high fidelity gate operations enabled the implementation of repetitive quantum error correction. Future applications towards analog and digital quantum simulations will be indicated and briefly discussed.