Ab initio quantum chemistry with neural-network wavefunctions

A non-Markovian neural quantum propagator and its application in the simulation of ultrafast nonlinear spectra

The accurate solution of dissipative quantum dynamics plays an important role in the simulation of open quantum systems. Here, we propose a machine learning-based universal solver for the hierarchical equations of motion, one of the most widely used approaches which takes into account non-Markovian effects and nonperturbative system-environment interactions in a numerically exact manner. We develop a neural quantum propagator model by utilizing the neural network architecture, which avoids time-consuming iterations and can be used to evolve any initial quantum state for arbitrarily long times. To demonstrate the efficacy of our model, we apply it to the simulation of population dynamics and linear and two-dimensional spectra of the Fenna-Matthews-Olson complex.

Spin-symmetry-enforced solution of the many-body Schrödinger equation with a deep neural network

The integration of deep neural networks with the variational Monte Carlo (VMC) method has marked a substantial advancement in solving the Schrödinger equation. In this work we enforce spin symmetry in the neural-network-based VMC calculation using a modified optimization target. Our method is designed to solve for the ground state and multiple excited states with target spin symmetry at a low computational cost. It predicts accurate energies while maintaining the correct symmetry in strongly correlated systems, even in cases in which different spin states are nearly degenerate. Our approach also excels at spin-gap calculations, including the singlet-triplet gap in biradical systems, which is of high interest in photochemistry. Overall, this work establishes a robust framework for efficiently calculating various quantum states with specific spin symmetry in correlated systems.

Accurate Calculation of Interatomic Forces with Neural Networks Based on a Generative Transformer Architecture

Using neural networks to express electronic wave functions represents a new paradigm for solving the Schrödinger equation in quantum chemistry. For practical quantum chemistry simulations, one needs to know not only energies of molecules, but also accurate forces acting on constituent atoms. In this work, we achieve the accurate calculation of interatomic forces on QiankunNet, a platform that combines transformer-based deep neural networks with efficient batched autoregressive sampling. Our approach permits the application of the Hellmann-Feynman theorem to force calculations without introducing corrective Pulay terms. The results show that the calculated interatomic forces are in close agreement with those derived from the full configuration interaction method, irrespective of whether the system is a simple molecule or a strongly correlated electron system like a linear hydrogen chain. Furthermore, the calculated interatomic forces are employed for atomic relaxation in the torsional rotation process of ethylene, and the energy barrier obtained from the scanned potential energy surface is in excellent agreement with the experiment. Our work contributes to the application of artificial intelligence to broader quantum chemistry simulations, such as modeling challenging chemical transformations where electron correlations are difficult to describe.

Gaussian processes for finite size extrapolation of many-body simulations

Key to being able to accurately model the properties of realistic materials is being able to predict their properties in the thermodynamic limit. Nevertheless, because most many-body electronic structure methods scale as a high-order polynomial, or even exponentially, with system size, directly simulating large systems in their thermodynamic limit rapidly becomes computationally intractable. As a result, researchers typically estimate the properties of large systems that approach the thermodynamic limit by extrapolating the properties of smaller, computationally-accessible systems based on relatively simple scaling expressions. In this work, we employ Gaussian processes to more accurately and efficiently extrapolate many-body simulations to their thermodynamic limit. We train our Gaussian processes on Smooth Overlap of Atomic Positions (SOAP) descriptors to extrapolate the energies of one-dimensional hydrogen chains obtained using two high-accuracy many-body methods: coupled cluster theory and Auxiliary Field Quantum Monte Carlo (AFQMC). In so doing, we show that Gaussian processes trained on relatively short 10-30-atom chains can predict the energies of both homogeneous and inhomogeneous hydrogen chains in their thermodynamic limit with sub-milliHartree accuracy. Unlike standard scaling expressions, our GPR-based approach is highly generalizable given representative training data and is not dependent on systems' geometries or dimensionality. This work highlights the potential for machine learning to correct for the finite size effects that routinely complicate the interpretation of finite size many-body simulations.

Spiers Memorial Lecture: Quantum chemistry, classical heuristics, and quantum advantage

We describe the problems of quantum chemistry, the intuition behind classical heuristic methods used to solve them, a conjectured form of the classical complexity of quantum chemistry problems, and the subsequent opportunities for quantum advantage. This article is written for both quantum chemists and quantum information theorists. In particular, we attempt to summarize the domain of quantum chemistry problems as well as the chemical intuition that is applied to solve them within concrete statements (such as a classical heuristic cost conjecture) in the hope that this may stimulate future analysis.

Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation

Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. We employ time-dependent Jastrow factors and backflow transformations, enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we employ the time-dependent variational Monte Carlo technique and introduce a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately describe the time evolution, providing insight into quantum dynamical effects in interacting electronic systems, beyond the capabilities of mean-field.

Variational quantum imaginary time evolution for matrix product state Ansatz with tests on transcorrelated Hamiltonians

The matrix product state (MPS) Ansatz offers a promising approach for finding the ground state of molecular Hamiltonians and solving quantum chemistry problems. Building on this concept, the proposed technique of quantum circuit MPS (QCMPS) enables the simulation of chemical systems using a relatively small number of qubits. In this study, we enhance the optimization performance of the QCMPS Ansatz by employing the variational quantum imaginary time evolution (VarQITE) approach. Guided by McLachlan's variational principle, the VarQITE method provides analytical metrics and gradients, resulting in improved convergence efficiency and robustness of the QCMPS. We validate these improvements numerically through simulations of H2, H4, and LiH molecules. In addition, given that VarQITE is applicable to non-Hermitian Hamiltonians, we evaluate its effectiveness in preparing the ground state of transcorrelated Hamiltonians. This approach yields energy estimates comparable to the complete basis set (CBS) limit while using even fewer qubits. In particular, we perform simulations of the beryllium atom and LiH molecule using only three qubits, maintaining high fidelity with the CBS ground state energy of these systems. This qubit reduction is achieved through the combined advantages of both the QCMPS Ansatz and transcorrelation. Our findings demonstrate the potential practicality of this quantum chemistry algorithm on near-term quantum devices.

Understanding covalency in molecular f-block compounds from the synergy of spectroscopy and quantum chemistry

One of the most intensely studied areas of f-block chemistry is the nature of the bonds between the f-element and another species, and in particular the role played by covalency. Computational quantum chemical methods have been at the forefront of this research for decades and have a particularly valuable role, given the radioactivity of the actinide series. The very strong agreement that has recently emerged between theory and the results of a range of spectroscopic techniques not only facilitates deeper insight into the experimental data, but it also provides confidence in the conclusions from the computational studies. These synergies are shining new light on the nature of the f element-other element bond.

Analytical approach to structural chemistry origins of mechanical, acoustical and thermal properties

Crystalline matters with periodically arranged atoms found wide applications in modern science and technology. To facilitate the design of new materials and the advancement of existing ones, accurate and efficient models without relying too much on known inputs for predicting the functionalities are essential. Here, we propose an analytical approach for such a purpose, with only the knowledge of the structural chemistry of crystals. Based on the electrostatic interaction between periodically arranged atoms, the 1st, 2nd and 3rd derivatives of interatomic potential, respectively, enable a prediction of ten kinds in total of mechanical, acoustical and thermal properties. Over a thousand measurements are collected from ∼500 literatures, this results in the symmetric mean percentage error (SMPE) within ±25% and the symmetric mean absolute percentage error (SMAPE) ranging from 22%∼74% across all properties predicted, which further enables a revelation of bond characteristics as the most important but implicit origin for functionalities.

An Improved Penalty-Based Excited-State Variational Monte Carlo Approach with Deep-Learning Ansatzes

We introduce several improvements to the penalty-based variational quantum Monte Carlo (VMC) algorithm for computing electronic excited states of Entwistle et al. [Nat. Commun. 14, 274 (2023)] and demonstrate that the accuracy of the updated method is competitive with other available excited-state VMC approaches. A theoretical comparison of the computational aspects of these algorithms is presented, where several benefits of the penalty-based method are identified. Our main contributions include an automatic mechanism for tuning the scale of the penalty terms, an updated form of the overlap penalty with proven convergence properties, and a new term that penalizes the spin of the wave function, enabling the selective computation of states with a given spin. With these improvements, along with the use of the latest self-attention-based ansatz, the penalty-based method achieves a mean absolute error below 1 kcal/mol for the vertical excitation energies of a set of 26 atoms and molecules, without relying on variance matching schemes. Considering excited states along the dissociation of the carbon dimer, the accuracy of the penalty-based method is on par with that of natural-excited-state (NES) VMC, while also providing results for additional sections of the potential energy surface, which were inaccessible with the NES method. Additionally, the accuracy of the penalty-based method is improved for a conical intersection of ethylene, with the predicted angle of the intersection agreeing well with both NES-VMC and multireference configuration interaction.

Accurate computation of quantum excited states with neural networks

We present an algorithm to estimate the excited states of a quantum system by variational Monte Carlo, which has no free parameters and requires no orthogonalization of the states, instead transforming the problem into that of finding the ground state of an expanded system. Arbitrary observables can be calculated, including off-diagonal expectations, such as the transition dipole moment. The method works particularly well with neural network ansätze, and by combining this method with the FermiNet and Psiformer ansätze, we can accurately recover excitation energies and oscillator strengths on a range of molecules. We achieve accurate vertical excitation energies on benzene-scale molecules, including challenging double excitations. Beyond the examples presented in this work, we expect that this technique will be of interest for atomic, nuclear, and condensed matter physics.

Second-order optimization strategies for neural network quantum states

The Variational Monte Carlo (VMC) method has recently seen important advances through the use of neural network quantum states. While more and more sophisticated ansatze have been designed to tackle a wide variety of quantum many-body problems, modest progress has been made on the associated optimization algorithms. In this work, we revisit the Kronecker-Factored Approximate Curvature (KFAC), an optimizer that has been used extensively in a variety of simulations. We suggest improvements in the scaling and the direction of this optimizer and find that they substantially increase its performance at a negligible additional cost. We also reformulate the VMC approach in a game theory framework, to propose a novel optimizer based on decision geometry. We find that on a practical test case for continuous systems, this new optimizer consistently outperforms any of the KFAC improvements in terms of stability, accuracy and speed of convergence. Beyond VMC, the versatility of this approach suggests that decision geometry could provide a solid foundation for accelerating a broad class of machine learning algorithms. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

In Silico Chemical Experiments in the Age of AI: From Quantum Chemistry to Machine Learning and Back

Computational chemistry is an indispensable tool for understanding molecules and predicting chemical properties. However, traditional computational methods face significant challenges due to the difficulty of solving the Schrödinger equations and the increasing computational cost with the size of the molecular system. In response, there has been a surge of interest in leveraging artificial intelligence (AI) and machine learning (ML) techniques to in silico experiments. Integrating AI and ML into computational chemistry increases the scalability and speed of the exploration of chemical space. However, challenges remain, particularly regarding the reproducibility and transferability of ML models. This review highlights the evolution of ML in learning from, complementing, or replacing traditional computational chemistry for energy and property predictions. Starting from models trained entirely on numerical data, a journey set forth toward the ideal model incorporating or learning the physical laws of quantum mechanics. This paper also reviews existing computational methods and ML models and their intertwining, outlines a roadmap for future research, and identifies areas for improvement and innovation. Ultimately, the goal is to develop AI architectures capable of predicting accurate and transferable solutions to the Schrödinger equation, thereby revolutionizing in silico experiments within chemistry and materials science.

Empowering deep neural quantum states through efficient optimization

Computing the ground state of interacting quantum matter is a long-standing challenge, especially for complex two-dimensional systems. Recent developments have highlighted the potential of neural quantum states to solve the quantum many-body problem by encoding the many-body wavefunction into artificial neural networks. However, this method has faced the critical limitation that existing optimization algorithms are not suitable for training modern large-scale deep network architectures. Here, we introduce a minimum-step stochastic-reconfiguration optimization algorithm, which allows us to train deep neural quantum states with up to 106 parameters. We demonstrate our method for paradigmatic frustrated spin-1/2 models on square and triangular lattices, for which our trained deep networks approach machine precision and yield improved variational energies compared to existing results. Equipped with our optimization algorithm, we find numerical evidence for gapless quantum-spin-liquid phases in the considered models, an open question to date. We present a method that captures the emergent complexity in quantum many-body problems through the expressive power of large-scale artificial neural networks.

Improved optimization for the neural-network quantum states and tests on the chromium dimer

The advent of Neural-network Quantum States (NQS) has significantly advanced wave function ansatz research, sparking a resurgence in orbital space variational Monte Carlo (VMC) exploration. This work introduces three algorithmic enhancements to reduce computational demands of VMC optimization using NQS: an adaptive learning rate algorithm, constrained optimization, and block optimization. We evaluate the refined algorithm on complex multireference bond stretches of H2O and N2 within the cc-pVDZ basis set and calculate the ground-state energy of the strongly correlated chromium dimer (Cr2) in the Ahlrichs SV basis set. Our results achieve superior accuracy compared to coupled cluster theory at a relatively modest CPU cost. This work demonstrates how to enhance optimization efficiency and robustness using these strategies, opening a new path to optimize large-scale restricted Boltzmann machine-based NQS more effectively and marking a substantial advancement in NQS's practical quantum chemistry applications.

A Vision for the Future of Multiscale Modeling

The rise of modern computer science enabled physical chemistry to make enormous progresses in understanding and harnessing natural and artificial phenomena. Nevertheless, despite the advances achieved over past decades, computational resources are still insufficient to thoroughly simulate extended systems from first principles. Indeed, countless biological, catalytic and photophysical processes require ab initio treatments to be properly described, but the breadth of length and time scales involved makes it practically unfeasible. A way to address these issues is to couple theories and algorithms working at different scales by dividing the system into domains treated at different levels of approximation, ranging from quantum mechanics to classical molecular dynamics, even including continuum electrodynamics. This approach is known as multiscale modeling and its use over the past 60 years has led to remarkable results. Considering the rapid advances in theory, algorithm design, and computing power, we believe multiscale modeling will massively grow into a dominant research methodology in the forthcoming years. Hereby we describe the main approaches developed within its realm, highlighting their achievements and current drawbacks, eventually proposing a plausible direction for future developments considering also the emergence of new computational techniques such as machine learning and quantum computing. We then discuss how advanced multiscale modeling methods could be exploited to address critical scientific challenges, focusing on the simulation of complex light-harvesting processes, such as natural photosynthesis. While doing so, we suggest a cutting-edge computational paradigm consisting in performing simultaneous multiscale calculations on a system allowing the various domains, treated with appropriate accuracy, to move and extend while they properly interact with each other. Although this vision is very ambitious, we believe the quick development of computer science will lead to both massive improvements and widespread use of these techniques, resulting in enormous progresses in physical chemistry and, eventually, in our society.

Electric Polarization from a Many-Body Neural Network Ansatz

Ab initio calculation of dielectric response with high-accuracy electronic structure methods is a long-standing problem, for which mean-field approaches are widely used and electron correlations are mostly treated via approximated functionals. Here we employ a neural network wave function ansatz combined with quantum Monte Carlo method to incorporate correlations into polarization calculations. On a variety of systems, including isolated atoms, one-dimensional chains, two-dimensional slabs, and three-dimensional cubes, the calculated results outperform conventional density functional theory and are consistent with the most accurate calculations and experimental data. Furthermore, we have studied the out-of-plane dielectric constant of bilayer graphene using our method and reestablished its thickness dependence. Overall, this approach provides a powerful tool to accurately describe electron correlation in the modern theory of polarization.

Artificial-Intelligence-Based Surrogate Solution of Dissipative Quantum Dynamics: Physics-Informed Reconstruction of the Universal Propagator

The accurate (or even approximate) solution of the equations that govern the dynamics of dissipative quantum systems remains a challenging task in quantum science. While several algorithms have been designed to solve those equations with different degrees of flexibility, they rely mainly on highly expensive iterative schemes. Most recently, deep neural networks have been used for quantum dynamics, but current architectures are highly dependent on the physics of the particular system and usually limited to population dynamics. Here we introduce an artificial-intelligence-based surrogate model that solves dissipative quantum dynamics by parametrizing quantum propagators as Fourier neural operators, which we train using both data set and physics-informed loss functions. Compared with conventional algorithms, our quantum neural propagator avoids time-consuming iterations and provides a universal superoperator that can be used to evolve any initial quantum state for arbitrarily long times. To illustrate the wide applicability of the approach, we employ our quantum neural propagator to compute the population dynamics and time-correlation functions of the Fenna-Matthews-Olson complex.

AI in computational chemistry through the lens of a decade-long journey

This article gives a perspective on the progress of AI tools in computational chemistry through the lens of the author's decade-long contributions put in the wider context of the trends in this rapidly expanding field. This progress over the last decade is tremendous: while a decade ago we had a glimpse of what was to come through many proof-of-concept studies, now we witness the emergence of many AI-based computational chemistry tools that are mature enough to make faster and more accurate simulations increasingly routine. Such simulations in turn allow us to validate and even revise experimental results, deepen our understanding of the physicochemical processes in nature, and design better materials, devices, and drugs. The rapid introduction of powerful AI tools gives rise to unique challenges and opportunities that are discussed in this article too.

A Nonstochastic Optimization Algorithm for Neural-Network Quantum States

Neural-network quantum states (NQS) employ artificial neural networks to encode many-body wave functions in a second quantization through variational Monte Carlo (VMC). They have recently been applied to accurately describe electronic wave functions of molecules and have shown the challenges in efficiency compared with traditional quantum chemistry methods. Here, we introduce a general nonstochastic optimization algorithm for NQS in chemical systems, which deterministically generates a selected set of important configurations simultaneously with energy evaluation of NQS. This method bypasses the need for Markov-chain Monte Carlo within the VMC framework, thereby accelerating the entire optimization process. Furthermore, this newly developed nonstochastic optimization algorithm for NQS offers comparable or superior accuracy compared to its stochastic counterpart and ensures more stable convergence. The application of this model to test molecules exhibiting strong electron correlations provides further insight into the performance of NQS in chemical systems and opens avenues for future enhancements.