Before revision:  Thus, our result has several direct applications, including polynomial-time algorithms 1)  to sample from  matrix Langevin distributions studied in statistics, 2)  to sample from continuous maximum entropy distributions with applications to quantum inference , and 3)  for differentially private low-rank approximation.
After revision:  Thus, our result has several direct applications, including polynomial-time algorithms 1) to sample from matrix Langevin distributions studied in statistics, 2) to sample from continuous max-entropy distributions on unitary orbits, which implies an efficient algorithm to sample a pure quantum state from the entropy-maximizing ensemble representing a given density matrix , and 3) for differentially private low-rank approximation.