Forming a heterointerface is a materials design strategy that can access an astronomically large phase space of systems rich in novel fundamental phenomena and device applications. However, the size and complexity of this phase space necessitate new theoretical and computational approaches for optimal interface design. In this dissertation I will present two efforts in which we developed theoretical frameworks for overcoming major obstacles in studying quantum materials heterostructures: (i) Mismatched INterface Theory (MINT), a cluster density-functional theory (DFT) method for studying the electronic structure of incommensurate interfaces from first principles, and (ii) InterMatch, a high-throughput data-driven code and materials database for interface design.