Usingourmethod weestablish anewbenchmark bycalculating the most accurate variational ground state energies ever published for a number of different atoms and molecules. Wesystematically break down and measure our improvements, focusing in particular on the effect of increasing physical prior knowledge.

Solving the quantum many-body Schrödinger 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 the Fisher--Rao metric, and corresponds to the probability mass, rather than it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.

Finding accurate solutions to the Schrödinger equation is the key unsolved challenge of computational chemistry. Given its importance for the development of new chemical compounds, decades of research have been dedicated to this problem, but due to the large dimensionality even the best available methods do not yet reach the desired accuracy.Recently the combination of deep learning with Monte Carlo methods has emerged as a promising way to obtain highly accurate energies and moderate scaling of computational cost. In this paper we significantly contribute towards this goal by introducing a novel deep-learning architecture that achieves 40-70% lower energy error at 6x lower computational cost compared to previous approaches. Using our method we establish a new benchmark by calculating the most accurate variational ground state energies ever published for a number of different atoms and molecules.We systematically break down and measure our improvements, focusing in particular on the effect of increasing physical prior knowledge.We surprisingly find that increasing the prior knowledge given to the architecture can actually decrease accuracy.

The proof, known to be so hard that a mathematician once offered 10 martinis to whoever could figure it out, uses number theory to explain quantum fractals. In 1974, five years before he wrote his Pulitzer Prize-winning book, Douglas Hofstadter was a graduate student in physics at the University of Oregon. When his doctoral adviser went on sabbatical to Regensburg, Germany, Hofstadter tagged along, hoping to practice his German. The pair joined a group of brilliant theoretical physicists who were agonizing over a particular problem in quantum theory. They wanted to determine the energy levels of an electron in a crystal grid placed near a magnet. Hofstadter was the odd one out, unable to follow the others' line of thought. "Part of my luck was that I couldn't keep up with them," he said.

Finding fast and accurate approaches to solving Schrödinger equations is a central challenge in quantum chemistry, with far-reaching implications for material science and pharmaceutical development. The ability to solve this equation precisely would unlock a plethora of properties inherent to the microscopic systems being studied.

More specifically, we propose "Wasserstein Quantum Monte Carlo"

The Quantum Schrödinger Bridge Problem (QSBP) describes the evolution of a stochastic process between two arbitrary probability distributions, where the dynamics are governed by the Schrödinger equation rather than by the traditional real-valued wave equation. Although the QSBP is known in the mathematical literature, we formulate it here from a Lagrangian perspective and derive its main features in a way that is particularly suited to generative modeling. We show that the resulting evolution equations involve the so-called Bohm (quantum) potential, representing a notion of non-locality in the stochastic process. This distinguishes the QSBP from classical stochastic dynamics and reflects a key characteristic typical of quantum mechanical systems. In this work, we derive exact closed-form solutions for the QSBP between Gaussian distributions. Our derivation is based on solving the Fokker-Planck Equation (FPE) and the Hamilton-Jacobi Equation (HJE) arising from the Lagrangian formulation of dynamical Optimal Transport. We find that, similar to the classical Schrödinger Bridge Problem, the solution to the QSBP between Gaussians is again a Gaussian process; however, the evolution of the covariance differs due to quantum effects. Leveraging these explicit solutions, we present a modified algorithm based on a Gaussian Mixture Model framework, and demonstrate its effectiveness across several experimental settings, including single-cell evolution data, image generation, molecular translation and applications in Mean-Field Games.

Finding accurate solutions to the Schrödinger equation is the key unsolved challenge of computational chemistry.