Kyprianidis, AntonisRichard Feynman in 1981 suggested using a quantum machine to simulate quantum mechanics.Peter Shor in 1994 showed that a quantum computer could factor numbers much more efficiently than a conventional one. Since then, the explosion of the quantum information field is attesting to how motivation and funding work miracles. Research labs in the field are multiplying, commercial companies manufacturing prototypes are proliferating, undergraduate Physics curricula incorporate more than one courses in aspects of quantum information, quantum advantage over classical computers has been claimed, and the United States and European Union will be spending more than \$$10^9$ each in quantum information over the next few years. Naturally, this expansion has led to diversification of the devices being developed. The quantum information systems that cannot simulate an arbitrary evolution, but are specialized in a specific set of Hamiltonians, are called quantum \emph{simulators}. They enjoy the luxury of being able to surpass computational abilities of classical computers \emph{right now}, at the expense of only doing so for a narrow type of problem. Among those systems, ions trapped in vacuum by electric fields and manipulated with light have proved to be a leading platform in emulating quantum magnetism models. In this thesis I present trapped-ion experiments realizing a prethermal discrete time crystal. This exotic phase occurs in non-equilibrium matter subject to an external periodic drive. Normally, the ensuing Floquet heating maximizes the system entropy, leaving us with a trivial, infinite-temperature state. However, we are able to parametrically slow down this heating by tuning the drive frequency. During the time window of slow thermalization, we define an order parameter and observe two different regimes, based on whether it spontaneously breaks the discrete time translation symmetry of the drive or it preserves it. Furthermore, I demonstrate a simple model of electric field noise classically heating an ion in an anharmonic confining potential. As ion traps shrink, this kind of noise may become more significant. And finally, I discuss a handful of error sources. As quantum simulation experiments progress to more qubits and complicated sequences, accounting for system imperfections is becoming an integral part of the process.enDissertationQuantum physicsApplied physicsEngineeringAMODecoherencePrethermal discrete time crystalsQuantum opticsQuantum simulationQuantum simulator