# Wasserstein Quantum Monte Carlo: A Novel Approach for Solving the Quantum Many-Body Schr"odinger Equation

arXiv: https://arxiv.org/abs/2307.07050 · HTML: https://mlsanity.com/paper/2307.07050/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Builds on (most cited first, 10 of 10)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)

## Built on by (most cited first, 12 of 12)

- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Wasserstein Policy Optimization](https://mlsanity.com/paper/2505.00663/) (2025)
- [Ab-initio variational wave functions for the time-dependent many-electron Schr"odinger equation](https://mlsanity.com/paper/2403.07447/) (2024)
- [Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations](https://mlsanity.com/paper/2404.01145/) (2024)
- [Functional Neural Wavefunction Optimization](https://mlsanity.com/paper/2507.10835/) (2025)
- [Diffusion-Augmented Markov Decision Processes for Maximum Entropy Reinforcement Learning](https://mlsanity.com/paper/2512.02019/) (2025)
- [nabla2DFT: A Universal Quantum Chemistry Dataset of Drug-Like Molecules and a Benchmark for Neural Network Potentials](https://mlsanity.com/paper/2406.14347/) (2024)
- [Spin-Adapted Neural Network Wavefunctions in Real Space](https://mlsanity.com/paper/2511.01671/) (2025)
- [Sorting Out Quantum Monte Carlo](https://mlsanity.com/paper/2311.05598/) (2023)
- [Efficient Numerical Algorithm for Large-Scale Damped Natural Gradient Descent](https://mlsanity.com/paper/2310.17556/) (2023)
- [Rethinking the Flow-Based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective](https://mlsanity.com/paper/2602.01179/) (2026)
- [Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspective](https://mlsanity.com/paper/2602.01182/) (2026)

## Abstract (arXiv)

Solving the quantum many-body Schr"odinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher-Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.