We introduce a framework for fermionic variational Monte Carlo (VMC) in which a continuous normalizing flow (CNF) refines a fixed antisymmetric base wavefunction. The flow is implemented as a permutation-equivariant neural ODE, a smooth, topology-preserving map that learns correlations not captured by the base; equivariance preserves the antisymmetry of the base, so the flow can in principle improve any antisymmetric ansatz that can be sampled efficiently. We demonstrate this using Slater and Jastrow-Slater bases, though more expressive choices are admissible. Exact samples from the flow's Born distribution are obtained by pushing pre-cached base samples through the forward ODE, requiring no Markov chain Monte Carlo (MCMC) at training time. The base samples are generated offline and reused across training batches and runs, decoupling sample generation from parameter optimization and enabling embarrassingly parallel training across multiple GPUs. We introduce three novel permutation-equivariant vector field architectures: Pairwise Deep Sets (PDS), FermiNet Vector Fields (FVF), and Pairwise Deep Sets Gradient (PDSG), each offering a different balance of expressivity and computational cost. We further introduce an augmented dynamics formulation for kinetic energy computation that co-evolves the required derivative quantities as ODE state variables, eliminating differentiation through the ODE trajectory and yielding significant reductions in wall-clock time and memory. Training runs on systems of harmonically trapped spinless electrons demonstrate ground-state energies below CISD reference values. Scaling experiments demonstrate near-ideal strong scaling from 1 to 128 NVIDIA A100s using 32 GPU nodes of NERSC's Perlmutter supercomputer for systems of up to $N = 48$ particles in three dimensions.
We introduce a framework for fermionic variational Monte Carlo (VMC) in which a continuous normalizing flow (CNF) refines a fixed antisymmetric base wavefunction. The flow is implemented as a permutation-equivariant neural ODE, a smooth, topology-preserving map that learns correlations not captured by the base; equivariance preserves the antisymmetry of the base, so the flow can in principle improve any antisymmetric ansatz that can be sampled efficiently. We demonstrate this using Slater and Jastrow-Slater bases, though more expressive choices are admissible. Exact samples from the flow's Born distribution are obtained by pushing pre-cached base samples through the forward ODE, requiring no Markov chain Monte Carlo (MCMC) at training time. The base samples are generated offline and reused across training batches and runs, decoupling sample generation from parameter optimization and enabling embarrassingly parallel training across multiple GPUs. We introduce three novel permutation-equivariant vector field architectures: Pairwise Deep Sets (PDS), FermiNet Vector Fields (FVF), and Pairwise Deep Sets Gradient (PDSG), each offering a different balance of expressivity and computational cost. We further introduce an augmented dynamics formulation for kinetic energy computation that co-evolves the required derivative quantities as ODE state variables, eliminating differentiation through the ODE trajectory and yielding significant reductions in wall-clock time and memory. Training runs on systems of harmonically trapped spinless electrons demonstrate ground-state energies below CISD reference values. Scaling experiments demonstrate near-ideal strong scaling from 1 to 128 NVIDIA A100s using 32 GPU nodes of NERSC's Perlmutter supercomputer for systems of up to N=48 particles in three dimensions.
A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature ("cusps") of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrodinger Equation.
A central problem in quantum mechanics involves solving the Electronic Schrödinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature (``cusps'') of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrödinger Equation.
O(N2) Universal Antisymmetry in Fermionic Neural Networks
Fermionic neural network (FermiNet) is a recently proposed wavefunction Ansatz, which is used in variational Monte Carlo (VMC) methods to solve the many-electron Schrödinger equation. FermiNet proposes permutation-equivariant architectures, on which a Slater determinant is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function given sufficient parameters. However, the asymptotic computational bottleneck comes from the Slater determinant, which scales with O(N3) for N electrons. In this paper, we substitute the Slater determinant with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to O(N2). We formally prove that the pairwise construction built upon permutation-equivariant architectures can universally represent any antisymmetric function. Besides, this universality can be achieved via continuous approximators when we aim to represent ground-state wavefunctions.
Understanding strongly correlated electron systems remains one of the central challenges in condensed matter physics. Finding ab initio descriptions of non-perturbative systems is of great importance for modeling carbon-based materials such as graphene or fullerenes. In this regime, computational methods become indispensable tools for performing simulations, making predictions, and ultimately comparing to experiments. The Hubbard model offers a minimal description of the dynamics of electrons in such materials. As exact methods are limited to small system sizes by the exponentially growing Hilbert space, larger-scale first-principle studies rely on numerical approaches. Alongside powerful Hamiltonian-based methods, the auxiliary-field description of the model offers a natural setting for Lagrangian-based Monte Carlo sampling. However, substantial challenges, such as lack of ergodicity and the numerical sign problem, arise in these approaches, leading to long autocorrelation times, biased results, and large statistical uncertainties. In recent years, generative models, especially normalizing flows, have been applied extensively to various theories to address shortcomings of traditional Monte Carlo-based algorithms, such as critical slowing down or topological freezing. While these approaches have shown promise in alleviating these issues, a poorly understood and unfavorable scaling of these methods has stalled their progress, and whether they can reach state-of-the-art system sizes in lattice quantum chromodynamics remains to be seen. This motivates the application of these methods to condensed matter systems, where physically relevant system sizes are reachable at substantially smaller scales. In this work, we study for the first time the application of generative models to the finite-temperature auxiliary-field Hubbard model. We begin by employing an equivariant normalizing flow that respects multiple symmetries inherent to the Boltzmann distribution, demonstrating a substantial improvement in sampling efficiency over Hybrid Monte Carlo and standard non-equivariant flows. We further develop Symmetry-Enforcing Stochastic Modulation (SESaMo), a more scalable algorithm capable of modeling both exact and broken symmetries in normalizing flows, and show that it achieves significantly higher effective sample sizes than traditional equivariant techniques at large system sizes. Equipped with SESaMo, we compute physical observables at half-filling in excellent agreement with exact diagonalization at substantially lower computational cost than Hybrid Monte Carlo. We then turn to the doped Hubbard model, where the theory exhibits a numerical sign problem. We introduce an annealing scheme to simulate the Hubbard model at finite chemical potential in the spin basis. Making use of the less severe sign problem in this description, the annealing scheme enables ergodic sampling of the multi-modal distribution, yielding more accurate results with substantially reduced uncertainties compared to state-of-the-art optimized Hybrid Monte Carlo. Lastly, we employ a masked autoencoder for distribution estimation to learn the Boltzmann distribution of the discrete auxiliary-field Hubbard model. Preliminary results show a significantly higher effective sample size compared to normalizing flows with continuous degrees of freedom, while also exhibiting a milder sign problem than the continuous spin basis.
An efficient and expressive wave function Ansatz is key to scalable solutions for complex many-body electronic structures. While Slater determinants are predominantly used for constructing antisymmetric electronic wave function Ansätze, this construction can result in limited expressiveness when the targeted wave function is highly complex. In this work, we introduce Waveflow, an innovative framework for learning many-body fermionic wave functions using boundary-conditioned normalizing flows. Instead of relying on Slater determinants, Waveflow imposes antisymmetry by defining the fundamental domain of the wave function and applying necessary boundary conditions. A key challenge in using normalizing flows for this purpose is addressing the topological mismatch between the prior and target distributions. We propose using O-spline priors and I-spline bijections to handle this mismatch, which allows for flexibility in the node number of the distribution while automatically maintaining its square-normalization property. We apply Waveflow to a one-dimensional many-electron system, where we variationally minimize the system’s energy using variational quantum Monte Carlo (VQMC). Our experiments demonstrate that Waveflow can effectively resolve topological mismatches and faithfully learn the ground-state wave function.
We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a compact domain can be approximated to arbitrary accuracy by a linear combination of K antisymmetric basis functions--such as pairwise products or Slater determinants--multiplied by symmetric functions. A key result is that the number of required bases is provably small: K=1 suffices in one-dimensional continua (and on lattices in any dimension), K=2 suffices in two dimensions, and in higher dimensions K grows at most linearly with particle number. The antisymmetric bases can be learned by small neural networks, while the symmetric factors are implemented by permutation-invariant networks whose width scales only linearly with particle number. Thus, Fermi Sets achieve universal approximation of fermionic wavefunctions with minimal overhead while retaining clear physical interpretability. As a numerical illustration, a single Fermi Sets model applied to metallic solid hydrogen in three dimensions, trained simultaneously across multiple nuclear geometries, surpasses all diffusion Monte Carlo benchmarks.