Sampling Matrices From Harish‐Chandra–Itzykson–Zuber Densities With Applications to Quantum Inference and Differential Privacy
Bibliographic record
Abstract
ABSTRACT Given two Hermitian matrices and , the Harish‐Chandra–Itzykson–Zuber (HCIZ) distribution is given by the density with respect to the Haar measure on the unitary group. Random unitary matrices distributed according to the HCIZ distribution are important in various settings in physics and random matrix theory, but the problem of sampling efficiently from this distribution has remained open. We present two algorithms to sample matrices from distributions that are close to the HCIZ distribution. The first algorithm produces samples that are ‐close in total variation, and the number of arithmetic operations required depends on . The second algorithm comes with a stronger guarantee that the samples are ‐close in infinity divergence; however, its number of arithmetic operations depends polynomially on . Our results have the following applications: (1) an efficient algorithm to sample from complex versions of matrix Langevin distributions studied in statistics, (2) an efficient algorithm to sample from continuous maximum entropy distributions over unitary orbits, which in turn implies an efficient algorithm to sample a pure quantum state from the entropy‐maximizing ensemble representing a given density matrix, and (3) an efficient algorithm for differentially private rank‐ approximation that comes with improved utility bounds for .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.026 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".