Before revision:  The main contribution of this paper is an algorithm to sample a matrix from a distribution that is \varepsilon-close to the given HCIZ density and whose running time depends polylogarithmically on 1/ \varepsilon. Interestingly, HCIZ densities can also be viewed as exponential densities on \mathrm{U (n)-orbits, and in this setting, they have been widely studied .
After revision:  The main contribution of this paper is an algorithm to sample a matrix from a distribution that is \varepsilon-close to the given HCIZ density and whose running time depends polylogarithmically on 1/ \xi, the number of bits needed to encode Y and \Lambda, and the differences of the largest and the smallest eigenvalues of Y and \Lambda. HCIZ densities can also be viewed as exponential densities on \mathrm{U (n)-orbits, and in this setting, they have been widely studied .