We introduce quantum subspace states for encoding arbitrary dimensional subspaces of n-dimensional space, generalizing encodings for vectors that had been previously used in quantum machine learning (QML). Subspace states are used to obtain three new QML algorithms: a determinant sampling algorithm with polynomial speedups over classical, a quantum SVD/SVT algorithm for compound matrices with potentially exponential speedups over classical, and a quantum topological data analysis (TDA) algorithm that uses logarithmic depth circuits improving upon previous constructions.