This paper proposes a new deterministic sampling strategy for constructing polynomial chaos approximations for expensive physics simulation models. The proposed approach, effectively subsampled quadratures involves sparsely subsampling an existing tensor grid using QR column pivoting. For polynomial interpolation using hyperbolic or total order sets, we then solve the following square least squares problem. For polynomial approximation, we use a column pruning heuristic that removes columns based on the highest total orders and then solves the tall least squares problem. While we provide bounds on the condition number of such tall submatrices, it is difficult to ascertain how column pruning effects solution accuracy as this is problem specific. We conclude with numerical experiments on an analytical function and a model piston problem that show the efficacy of our approach compared with randomized subsampling. We also show an example where this method fails.