A novel matrix completion problem is considered herein: observations based on fully sampled columns and quasi-polynomial side information is exploited. The framework is motivated by quantum chemistry problems wherein full matrix computation is expensive, but partial computations only lead to column information. The proposed algorithm successfully estimates the row-space of a true matrix given a priori knowledge of the true matrix. A theoretical error bound is provided, which captures the possible inaccuracies of the side information. This work designs the first provable matrix approximation algorithm using just column samples. The proposed algorithm is validated via simulations which enable the characterization of the amount of information provided by the quasi-polynomial side information.