The field of experimental quantum information processing is already at the stage where few-qubit quantum computers are available, such as the one in our laboratory, consisting of a string of 40Ca+ ions confined in a macroscopic linear Paul trap. In this work, improvements to the experimental setup are shown, followed by recent experiments. A new software for compiling quantum algorithms into experimental pulse sequences is described, that improves on previously existing tools. Then, a new laser setup for Raman cooling is shown. These tools are applied to experiments exploring two research lines: quantum computation and quantum simulations. The first experiment reported is a scalable implementation of Shor's algorithm for integer factoring, paradigmatic for quantum computation. The second experiment is a quantum simulation of quantum electrodynamics, as a particular case of lattice gauge theories, which are fundamental for high-energy physics. Here it is experimentally demonstrated how these theories can be efficiently simulated on a quantum computer.