Schr\"{o}dinger Generator for High-Dimensional Integration and Sampling on Quantum Many-Body States

Abstract

Integration and sampling in high dimensions are among central challenges in modern science and technology, underlying applications ranging from quantum many-body physics to Bayesian inference and artificial intelligence. Although conventional Monte Carlo methods are formally scalable, their efficiency deteriorates rapidly in the presence of strong correlations or sharp features in high-dimensional configuration space. Here we introduce a new framework, termed the Schr"odinger Generator, for integration and sampling based on the explicit optimization of coordinate transformations. The method decomposes the total Jacobian into two complementary components, including an adaptive map that minimizes estimator variance by learning the marginal structure in each dimension, and a normalizing-flow-based transformation that captures non-factorizable correlations in the target distribution. A final resampling step guarantees unbiased sampling even when the learned transformation is imperfect. We demonstrate stable and scalable performance for nuclear quantum many-body states in dimensions exceeding 600. Short-range correlations among nucleons in finite nucleus are faithfully reproduced. The framework offers a physically transparent approach to high-dimensional stochastic integration and sampling, opening new possibilities for simulations of complex quantum systems.