  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.
