All known simulations of quantum electronic systems on classical computers require resources that scale exponentially with the number of electrons. Quantum computers offer a computationally-efficient alternative. Various algorithms have been developed to attempt to solve for the ground state energy of a molecule on quantum computers, both in mapping the problem to quantum computers and in solving the problem using quantum simulation. In this thesis, we simulate the hydrogen molecule on the open-access quantum computers built by IBM and Rigetti using the phase estimation algorithm (PEA). We demonstrate the limitations of current quantum computers for making useful quantum chemistry simulations using PEA and provide a framework for testing open source quantum computers as their technology improves over time.