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Given two $n \times n$ Hermitian matrices $Y$ and $\Lambda$, theHarish-Chandra-Itzykson-Zuber (HCIZ) distribution on the unitary group$\text{U}(n)$ is $e^{\text{tr}(U\Lambda U^*Y)}d\mu(U)$, where $\mu$ is the Haarmeasure on $\text{U}(n)$. The density $e^{\text{tr}(U\Lambda U^*Y)}$ is knownas the HCIZ density. Random unitary matrices distributed according to the HCIZdensity are important in various settings in physics and random matrix theory.However, the basic question of efficient sampling from the HCIZ distributionhas remained open. We present two efficient algorithms to sample matrices fromdistributions that are close to the HCIZ distribution. The first algorithmoutputs samples that are $\xi$-close in total variation distance and requirespolynomially many arithmetic operations in $\log 1/\xi$ and the number of bitsneeded to encode $Y$ and $\Lambda$. The second algorithm comes with a strongerguarantee that the samples are $\xi$-close in infinity divergence, but thenumber of arithmetic operations depends polynomially on $1/\xi$, the number ofbits needed to encode $Y$ and $\Lambda$, and the differences of the largest andthe smallest eigenvalues of $Y$ and $\Lambda$.  HCIZ densities can also be viewed as exponential densities on$\text{U}(n)$-orbits, and these densities have been studied in statistics,machine learning, and theoretical computer science. Thus our results have thefollowing applications: 1) an efficient algorithm to sample from complexversions of matrix Langevin distributions studied in statistics, 2) anefficient algorithm to sample from continuous max-entropy distributions onunitary orbits, which implies an efficient algorithm to sample a pure quantumstate from the entropy-maximizing ensemble representing a given density matrix,and 3) an efficient algorithm for differentially private rank-$k$approximation, with improved utility bounds for $k>1$.

Sampling Matrices from Harish-Chandra-Itzykson-Zuber Densities with Applications to Quantum Inference and Differential Privacy