The study of quantum computers and quantum algorithms has captured the attention of physics globally in recent decades. While some researchers are working towards constructing fault-tolerant large-scale quantum computers, others are theorizing about the capabilities of these new computers and developing quantum algorithms to leverage their increased abilities. This dissertation engages with the second area of research concerning quantum algorithms. The first part of this dissertation discusses quantum algorithms for state preparation, with a focus on thermal state preparation. A selection of other state preparation methods is briefly summarized before a novel method of thermal state preparation is laid out wherein a so-called heat pump is employed in a technique of active cooling. The second part of this dissertation discusses quantum algorithms that aim to address the problem of gauge violation during the simulation of lattice gauge theories. The problem of gauge violation in both quantum analog and digital simulations is presented before two related methods are delineated. These methods leverage the quantum Zeno effect with frequent measurements on a system's physical subspace to keep the system's state physical throughout the course of a simulation. Furthermore, the properties of gauge transformations are utilized to help curtail the growth of unwanted unphysical amplitudes.