Since quantum computation was first proposed, a number of quantum algorithms have been found that require exponentially fewer operations than the best known corresponding algorithms for conventional computers. Thus far, it is not clear that any of the algorithms yet proposed provide an exponential advantage for ubiquitous tasks performed in general purpose computing. In this talk I will present results showing progress on two fronts towards the goal of finding a killer application for quantum computers. I will present a new approach for simulating quantum systems on quantum computers that is more efficient than existing methods that are based on Trotter-Suzuki formulas. Next, I will show how quantum simulation algorithms, such as the algorithm I present, can be adapted to efficiently solve certain least squares fitting problems. I will then conclude discussing whether either of these applications truly is a killer application for quantum computation and discuss ongoing work that builds upon these results.