Ab initio molecular simulation provides a sub-atomic understanding of chemical reactions and properties. However, due to the exponential scaling of many-body simulations, one can only obtain approximate solutions to the electronic Schrodinger equation. Although, in principle one can increase the sophistication of an approximation arbitrarily until a desired accuracy is reached, more sophisticated calculations rapidly become intractable for even the largest supercomputers. Quantum computers provide a promising route to bypass these limitations in molecular simulation. However, current and near term quantum devices suffer from environmental noise and gate errors, thus limiting the calculations able to be achieved. In this talk, I will discuss basic ideas concerning the simulation of molecules on quantum devices, and describe two different algorithmic directions our collaborative team has recently devised for trying to extract the most computational work out of noisy devices.