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Solving the quantum many-body Schr\"odinger equation is a fundamental andchallenging problem in the fields of quantum physics, quantum chemistry, andmaterial sciences. One of the common computational approaches to this problemis Quantum Variational Monte Carlo (QVMC), in which ground-state solutions areobtained by minimizing the energy of the system within a restricted family ofparameterized wave functions. Deep learning methods partially address thelimitations of traditional QVMC by representing a rich family of wave functionsin terms of neural networks. However, the optimization objective in QVMCremains notoriously hard to minimize and requires second-order optimizationmethods such as natural gradient. In this paper, we first reformulate energyfunctional minimization in the space of Born distributions corresponding toparticle-permutation (anti-)symmetric wave functions, rather than the space ofwave functions. We then interpret QVMC as the Fisher-Rao gradient flow in thisdistributional space, followed by a projection step onto the variationalmanifold. This perspective provides us with a principled framework to derivenew QMC algorithms, by endowing the distributional space with better metrics,and following the projected gradient flow induced by those metrics. Morespecifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which usesthe gradient flow induced by the Wasserstein metric, rather than Fisher-Raometric, and corresponds to transporting the probability mass, rather thanteleporting it. We demonstrate empirically that the dynamics of WQMC results infaster convergence to the ground state of molecular systems.

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