Density functional theory is one of the most efficient and widely used computational methods of quantum mechanics, especially in fields such as solid state physics and quantum chemistry. From the theoretical perspecive, its central object is the universal density functional which contains all intrinsic information about the quantum system in question. Once the external potential is provided, in principle one can obtain the exact ground-state energy via a simple minimization. However, the universal density functional is a very complicated mathematical object and almost always it is replaced with its approximate variants. So far, no ``first principles'', mathematically consistent and convergent approximation procedure has been devised that has general applicability. In this paper, we propose such a procedure by first applying Moreau--Yosida regularization to make the exact functionals continuous (even differentiable) and then approximate the regularized functional by a neural network. The resulting neural network preserves the positivity and convexity of the exact functionals. More importantly, it is differentiable, so it can be directly used in a Kohn--Sham calculation.
Density functional theory is one of the most efficient and widely used computational methods of quantum mechanics, especially in fields such as solid state physics and quantum chemistry. From the theoretical perspecive, its central object is the universal density functional which contains all intrinsic information about the quantum system in question. Once the external potential is provided, in principle one can obtain the exact ground-state energy via a simple minimization. However, the universal density functional is a very complicated mathematical object and almost always it is replaced with its approximate variants. So far, no ``first principles'', mathematically consistent and convergent approximation procedure has been devised that has general applicability. In this paper, we propose such a procedure by first applying Moreau--Yosida regularization to make the exact functionals continuous (even differentiable) and then approximate the regularized functional by a neural network. The resulting neural network preserves the positivity and convexity of the exact functionals. More importantly, it is differentiable, so it can be directly used in a Kohn--Sham calculation.
Density functional theory underlies the most successful and widely used numerical methods for electronic structure prediction of solids. However, it has the fundamental shortcoming that the universal density functional is unknown. In addition, the computational result—energy and charge density distribution of the ground state—is useful for electronic properties of solids mostly when reduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can help to free density functional theory from these limitations. We study a theory of spinless fermions on a one-dimensional lattice. The density functional is implicitly represented by a neural network, which predicts, besides the ground-state energy and density distribution, density-density correlation functions. At no point do we require a band structure interpretation. The training data, obtained via exact diagonalization, feeds into a learning scheme inspired by active learning, which minimizes the computational costs for data generation. We show that the network results are of high quantitative accuracy and, despite learning on random potentials, capture both symmetry-breaking and topological phase transitions correctly.
Density functional theory underlies the most successful and widely used numerical methods for electronic structure prediction of solids. However, it has the fundamental shortcoming that the universal density functional is unknown. In addition, the computational result---energy and charge density distribution of the ground state---is useful for electronic properties of solids mostly when reduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate how machine learning algorithms can help to free density functional theory from these limitations. We study a theory of spinless fermions on a one-dimensional lattice. The density functional is implicitly represented by a neural network, which predicts, besides the ground-state energy and density distribution, density-density correlation functions. At no point do we require a band structure interpretation. The training data, obtained via exact diagonalization, feeds into an active learning scheme which minimizes the computational costs for data generation. We show that the network results are of high quantitative accuracy and, despite learning on random potentials, capture both symmetry-breaking and topological phase transitions correctly.
Kohn-Sham density functional theory (KS-DFT) provides the exact ground-state energy and electron density of a molecule, contingent on the as-yet-unknown universal exchange-correlation (XC) functional. Recent research has demonstrated that neural networks can efficiently learn to represent approximations to that functional, offering accurate generalizations to molecules not present during the training process. With the latest advancements in quantum-enhanced machine learning (ML), evidence is growing that quantum neural network (QNN) models may offer advantages in ML applications. In this work, we explore the use of QNNs for representing XC functionals, enhancing and comparing them to classical ML techniques. We present QNNs based on differentiable quantum circuits as quantum (hybrid) models for XC in KS-DFT, implemented across various architectures. We assess their performance on one-dimensional (1D) and 3D systems. To that end, we expand existing differentiable KS-DFT frameworks and propose strategies for efficient training of such functionals, highlighting the importance of fractional orbital occupation for accurate results. We prove that, under specific assumptions, noise-induced errors do not accumulate across SCF iterations, and we confirm this result through numerical experiments. Our best QNN-based XC functional yields energy profiles of the H 2 and planar H 4 molecules that deviate by no more than 1 mHa from the reference DMRG and FCI/6-31G results, respectively. Moreover, they reach chemical precision on a system, H 2 H 2 , not present in the training dataset, using only a few variational parameters. This work lays the foundation for the integration of quantum models in KS-DFT, thereby opening new avenues for expressing XC functionals in a differentiable way and facilitating computations of various properties.
The solution of complex many-body lattice models can often be found by defining an energy functional of the relevant density of the problem. For instance, in the case of the Hubbard model the spin-resolved site occupation is enough to describe the system's total energy. Similarly to standard density functional theory, however, the exact functional is unknown, and suitable approximations need to be formulated. By using a deep-learning neural network trained on exact-diagonalization results, we demonstrate that one can construct an exact functional for the Hubbard model. In particular, we show that the neural network returns a ground-state energy numerically indistinguishable from that obtained by exact diagonalization and, most importantly, that the functional satisfies the two Hohenberg-Kohn theorems: for a given ground-state density it yields the external potential, and it is fully variational in the site occupation.
We present a novel approach that combines machine learning with direct variational energy optimization via the density matrix to solve the Kohn-Sham equation in density functional theory. Instead of relying on the conventional self-consistent field method, our approach directly optimizes the ground state by predicting the density matrix using a neural network, thereby bypassing Hamiltonian matrix diagonalization. Our model employs equivariant neural networks to generate a physically constrained density matrix, enabling stable and efficient energy minimization. This method integrates the construction of training sets directly into the model training process and achieves high accuracy in predicting ground-state properties across various molecular and extended systems, thereby establishing a powerful machine learning paradigm for electronic structure optimization and paving the way for large-scale quantum simulations.
The field of quantum chemistry has long been defined by the central computational challenge of the quantum many-body fermionic system problem, main the electronic Schrödinger equation. This exponential scaling makes it computationally intractable to solve, the exact governing electronic Schrödinger equation. Approximation methods have been developed, ranging from the widely used Density Functional Theory (DFT) to Coupled Cluster (CC) theory. While powerful, these conventional approaches often involve controlled truncations which caused a trade-off between accuracy, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign oscillation problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). 1 Traditional Ansätze are also frequently engineered specifically for particular systems, which results in a fragmentation of results and fundamentally limits the transferability. Recent research demonstrates that with carefully designed architectures, ANNs can not only represent ground states of model systems with high precision but also tackle the sign problem, offering a promising path forward for strongly correlated electronic structure calculations.2,3 This review charts the paradigm shift driven by Neural Network Quantum States (NNQS), which leverage the representational power of deep learning to overcome these barriers. We detail the architectural evolution from Restricted Boltzmann Machines to autoregressive models like RNNs and Transformers, which enable exact, uncorrelated sampling and bypass critical bottlenecks.2,4,5Furthermore, we explore the geometric foundation of high-efficiency optimization via the neural networks, which decouples optimization cost from model complexity.6 The frontier of the field is marked by the integration of operator learning and hybrid quantum-classical frameworks, such as Reinforcement Learning for contracted quantum eigen-solvers and Quantum-Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware-efficient Ansätze.7,8 By framing the problem through the lens of computational resource allocation—specifically, trading exponential memory demands for polynomialcomplexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of those methods not merely progress into a powerful simulation-tools, but also a hybrid computational interface.9 This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.