Quantum computers hold the promise to solve certain problems exponentially faster than their classical counterparts. Trapped atomic ions are among the physical systems in which building such a computing device seems viable. In this work we present a small-scale quantum information processor based on a string of 40Ca+ ions confined in a macroscopic linear Paul trap.In this talk, I will discuss two recent experimental results, one is quantum chemistry simulation and the second is observation of dynamic phase transtion. In the first one, we implement a parametrized, gate-based sequence corresponding to an approximated Hamiltonian of two simple molecules - molecular hydrogen (H2) and lithium hydride (LiH) and determine the potential energy curve describing the molecular bond between the constituent atoms. In the second one, we present the direct observation of the realtime non-analyticities at Dynamical Phase transitions in a system of trapped ions simulating long-range transverse-field Ising models.