Quantum computation has the potential to dramatically speed up the solution to a number of important problems in physics, chemistry, and mathematics. Unfortunately, many quantum algorithms are infeasible on current and near-term quantum devices. In this talk, I will show how combining classical and quantum resources allows one to maximally utilize available quantum resources in the solution of eigenvalue problems, with a particular emphases on quantum chemical applications. Moreover, this approach will help to highlight the origin of a potential quantum speedup over classical calculations. The talk will be designed to appeal to a broad mathematical audience and will not assume an extensive background in quantum mechanics.