Artificial neural networks (ANNs) for material modeling have received significant interest. We recently reported an adaptation of ANNs based on Boltzmann machine (BM) architectures to an ansatz of the multiconfigurational many-electron wavefunction, designated neural-network quantum state (NQS), for quantum chemistry calculations. Here, this study presents its extended formalism to a quantum algorithm that enables the preparation of the NQS through quantum gates. The descriptors of the ANN model, which are chosen as occupancies of electronic configurations, are quantum-mechanically represented by qubits. Our algorithm may thus bring potential advantages over classical sampling-based computation employed in the previous studies. The NQS can be accurately formed through the quantum-native procedures, but the training of the model in terms of energy minimization is performed on a classical computer; thus, our approach is a class of variational quantum eigensolver. The BM models are related to the Gibbs distribution, and our preparation procedures exploit techniques of quantum phase estimation but with no Hamiltonian evolution. The proposed algorithm is assessed by implementing it on a quantum computer simulator. Illustrative molecular calculations at the complete-active-space configuration interaction level of theory are displayed, confirming consistency with the accuracy of our previous classical approaches.
Artificial neural networks (ANNs) for material modeling have received significant interest. We recently reported an adaptation of ANNs based on Boltzmann machine (BM) architectures to an ansatz of the multiconfigurational many-electron wavefunction, designated neural-network quantum state (NQS), for quantum chemistry calculations. Here, this study presents its extended formalism to a quantum algorithm that enables the preparation of the NQS through quantum gates. The descriptors of the ANN model, which are chosen as occupancies of electronic configurations, are quantum-mechanically represented by qubits. Our algorithm may thus bring potential advantages over classical sampling-based computation employed in the previous studies. The NQS can be accurately formed through the quantum-native procedures, but the training of the model in terms of energy minimization is performed on a classical computer; thus, our approach is a class of variational quantum eigensolver. The BM models are related to the Gibbs distribution, and our preparation procedures exploit techniques of quantum phase estimation but with no Hamiltonian evolution. The proposed algorithm is assessed by implementing it on a quantum computer simulator. Illustrative molecular calculations at the complete-active-space configuration interaction level of theory are displayed, confirming consistency with the accuracy of our previous classical approaches.
We use artificial neural networks (ANNs) based on the Boltzmann machine (BM) architectures as an encoder of ab initio molecular many-electron wave functions represented with the complete active space configuration interaction (CAS-CI) model. As first introduced by the work of Carleo and Troyer for physical systems, the coefficients of the electronic configurations in the CI expansion are parametrized with the BMs as a function of their occupancies that act as descriptors. This ANN-based wave function ansatz is referred to as the neural-network quantum state (NQS). The machine learning is used for training the BMs in terms of finding a variationally optimal form of the ground-state wave function on the basis of the energy minimization. It is relevant to reinforcement learning and does not use any reference data nor prior knowledge of the wave function, while the Hamiltonian is given based on a user-specified chemical structure in the first-principles manner. Carleo and Troyer used the restricted Boltzmann machine (RBM), which has hidden units, for the neural network architecture of NQS, while, in this study, we further introduce its replacement with the BM that has only visible units but with different orders of connectivity. For this hidden-node free BM, the second- and third-order BMs based on quadratic and cubic energy functions, respectively, were implemented. We denote these second- and third-order BMs as BM2 and BM3, respectively. The pilot implementation of the NQS solver into an exact diagonalization module of the quantum chemistry program was made to assess the capability of variants of the BM-based NQS. The test calculations were performed by determining the CAS-CI wave functions of illustrative molecular systems, indocyanine green, and dinitrogen dissociation. The simulated energies have been shown to converge to CAS-CI energy in most cases by improving RBM with an increasing number of hidden nodes. BM3 systematically yields lower energies than BM2, reproducing the CAS-CI energies of dinitrogen across potential energy curves within an error of 50 μE h .
ABSTRACT The field of quantum chemistry has long been defined by the central computational challenge of the quantum many‐body fermionic system problem—namely, the electronic Schrödinger equation. The exponential scaling of this equation makes it computationally intractable to solve exactly. As a result, various 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 cause a trade‐off between accuracy and computational cost, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). Traditional ansatz are also frequently engineered specifically for particular systems, which results in a fragmentation of methodologies and fundamentally limits their 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. 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. The high‐efficiency optimization via neural networks was also explored, which decouples optimization cost from model complexity. Furthermore, the frontier of the field is marked by the integration of operator learning and hybrid quantum‐classical frameworks, such as Reinforcement Learning for contractive quantum eigen‐solvers to generate quantum circuits for simulating many‐body molecular systems, and Quantum‐Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware‐efficient ansatz. By framing the problem through the lens of computational resource allocation and quantum hardware integration—specifically, trading exponential memory demands for polynomial‐complexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of these methods represents not merely progress toward powerful simulation tools, but also the creation of a hybrid computational interface. This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.
A longstanding computational challenge is the accurate simulation of many-body particle systems. Especially for deriving key characteristics of high-impact but complex systems such as battery materials and high entropy alloys (HEA). While simple models allow for simulations of the required scale, these methods often fail to capture the complex dynamics that determine the characteristics. A long-theorized approach is to use quantum computers for this purpose, which allows for a more efficient encoding of quantum mechanical systems. In recent years, the field of quantum computing has become significantly more mature. Furthermore, the rise in integration of machine learning with quantum computing further pushes to a near-term advantage. In this work we aim to improve the well-established quantum computing method for calculating the inter-atomic potential, the variational quantum eigensolver, by presenting an auto-encoded VQE with neural-network predictions: NN-AE-VQE. We apply a quantum autoencoder for a compressed quantum state representation of the atomic system, to which a naive circuit ansatz is applied. This reduces the number of circuit parameters to optimize, while still minimal reduction in accuracy. Additionally, we train a classical neural network to predict the circuit parameters to avoid computationally expensive parameter optimization. We demonstrate these methods on a H2 molecule, achieving chemical accuracy. We believe this method shows promise of efficiently capturing highly accurate systems while omitting current bottlenecks of variational quantum algorithms. Finally, we explore options for exploiting the algorithm structure and further algorithm improvements.
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.