Tight simulation of a distribution using conditional samples

We present an algorithm for simulating a distribution using prefix conditional samples (Adar, Fischer and Levi, 2024), as well as ``prefix-compatible'' conditional models such as the interval model (Cannone, Ron and Servedio, 2015) and the subcube model (CRS15, Bhattacharyya and Chakraborty, 2018). The sample complexity is $O(\log^2 N / \varepsilon^2)$ prefix conditional samples per query, which improves on the previously known $\tilde{O}(\log^3 N / \varepsilon^2)$ (Kumar, Meel and Pote, 2025). Moreover, our simulating distribution is $O(\varepsilon^2)$-close to the input distribution with respect to the Kullback-Leibler divergence, which is stricter than the usual guarantee of being $O(\varepsilon)$-close with respect to the total-variation distance. We show that our algorithm is tight with respect to the highly-related task of estimation: every algorithm that is able to estimate the mass of individual elements within $(1 \pm \varepsilon)$-multiplicative error must make $\Omega(\log^2 N / \varepsilon^2)$ prefix conditional samples per element.