Before revision:  Interestingly, HCIZ densities can also be viewed as exponential densities on \mathrm{U (n)-orbits, and in this setting, they have been widely studied . 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:  Interestingly, HCIZ densities can also be viewed as exponential densities on \mathrm{U (n)-orbits, and these densities have been studied in statistics, machine learning, and theoretical computer science. Thus our results have the following applications: 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.