Quantum neural functionals (QNFs) are introduced as a hybrid quantum-classical framework to advance density functional theory (DFT). QNFs employ quantum neural networks to capture nonlocal many-body correlations in exchange-correlation functionals. Symmetry-aware inductive biases, including rotational invariance under discrete O(3) subgroups and exact functional constraints, may enhance trainability and generalization while mitigating barren plateaus. Benchmark studies on diatomic molecules demonstrate that QNFs accurately reproduce binding energies and effectively describe strongly correlated regimes, such as H2 dissociation, outperforming both classical neural functionals and naive quantum models. These results highlight QNFs as a scalable and physically grounded approach for next-generation density functional approximations
Quantum neural functionals (QNFs) are introduced as a hybrid quantum-classical framework to advance density functional theory (DFT). QNFs employ quantum neural networks to capture nonlocal many-body correlations in exchange-correlation functionals. Symmetry-aware inductive biases, including rotational invariance under discrete O(3) subgroups and exact functional constraints, may enhance trainability and generalization while mitigating barren plateaus. Benchmark studies on diatomic molecules demonstrate that QNFs accurately reproduce binding energies and effectively describe strongly correlated regimes, such as H2 dissociation, outperforming both classical neural functionals and naive quantum models. These results highlight QNFs as a scalable and physically grounded approach for next-generation density functional approximations
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.
Density functional theory (DFT) is routinely employed in material science and quantum chemistry to simulate weakly correlated electronic systems. Recently, deep learning (DL) techniques have been adopted to develop promising functionals for the strongly correlated regime. DFT can be applied to quantum spin models too, but functionals based on DL have not been developed yet. Here, we investigate DL-based DFTs for random quantum Ising chains, both with nearest-neighbor and up to next-nearest-neighbor couplings. Our neural functionals are trained and tested on data produced via the Jordan-Wigner transformation, exact diagonalization, and tensor-network methods. An economical gradient-descent optimization is used to find the ground-state properties of previously unseen Hamiltonian instances. Notably, our nonlocal functionals drastically improve upon the common local density approximations, and they are designed to be scalable, allowing us to simulate chain sizes much larger than those used for training. The prediction accuracy is analyzed, paying attention to the spatial correlations of the learned functionals and to the role of quantum criticality. Our findings indicate a suitable strategy to extend the reach of other computational methods with a controllable accuracy.
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.
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.
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.
Quantum machine learning holds the promise of combining the success of classical machine learning methods with the power of quantum computing, however one of the largest obstacles facing the field is the problem of barren plateaus. Parameterised quantum circuits offer a flexible framework for developing quantum machine learning models, but their practicality is constrained by a trade-off between trainability and classical simulability. In general, circuits that are sufficiently expressive to model complex behaviour often exhibit barren plateaus, where gradients vanish and optimisation fails.
In this work we investigate a compositional approach to mitigate this trade-off by assembling larger quantum models from smaller subcomponents. To ensure trainability of these subcomponents, we describe a framework for constructing group-invariant loss functions, which introduce symmetry-induced inductive bias and lead to improved gradient behaviour and generalisation. In particular, we use this framework to design permutation-equivariant quantum graph neural networks for identifying maximal cliques in graphs. The models we construct exhibit superior training gradients through symmetry-induced bias, and our experiments demonstrate that the trained models generalise to larger, more complex problem instances.
Finally, inspired by Quantum-Informed Recursive Optimisation Algorithms (arXiv:2308.13607), we implement a recursive hybrid quantum-classical heuristic using the learned quantum models to guide a classical search procedure, demonstrating improved inference accuracy and scalability. Together, these results suggest that compositional circuits could be a viable pathway towards scalable quantum learning models that remain challenging to reproduce classically.