Abstract: Time evolution of quantum systems is of interest in physics, in chemistry, and, more recently, in computer science. Quantum computers are suggested as one route to propagating quantum systems far more efficiently than ordinary numerical methods.  In the past few years, researchers have actively been improving quantum simulation algorithms and I'll describe the contributions I've made during my postdoctoral work to advance the state-of-the-art by invoking group theoretic ideas. I will present algorithms for simulating spin-free fermionic wave functions on quantum computers. As a coda, I will also report on active and planned investigations into the computational nature of simulation.