Flow Perturbation++: Multi-Step Unbiased Jacobian Estimation for High-Dimensional Boltzmann Sampling

The scalability of continuous normalizing flows (CNFs) for unbiased Boltzmann sampling remains limited in high-dimensional systems due to the cost of Jacobian-determinant evaluation, which requires $D$ backpropagation passes through the flow layers. Existing stochastic Jacobian estimators such as the Hutchinson trace estimator reduce computation but introduce bias, while the recently proposed Flow Perturbation method is unbiased yet suffers from high variance. We present **Flow Perturbation++**, a variance-reduced extension of Flow Perturbation that discretizes the probability-flow ODE and performs unbiased stepwise Jacobian estimation at each integration step. This multi-step construction retains the unbiasedness of Flow Perturbation while achieves substantially lower estimator variance. Integrated into a Sequential Monte Carlo framework, Flow Perturbation++ achieves significantly improved equilibrium sampling on a 1000D Gaussian Mixture Model and the all-atom Chignolin protein compared with Hutchinson-based and single-step Flow Perturbation baselines.