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.
After revision:  We present two efficient algorithms to sample matrices from distributions that are close to the HCIZ distribution. The first algorithm outputs samples that are \xi-close in total variation distance and requires polynomially many arithmetic operations in \log 1/\xi and the number of bits needed to encode Y and \Lambda. The second algorithm comes with a stronger guarantee that the samples are \xi-close in infinity divergence, but the number of arithmetic operations depends polynomially on 1/ \varepsilon.