Quantum Monte Carlo coupled with neural network wavefunctions has shown success in computing ground states of quantum many-body systems. Existing optimization approaches compute the energy by sampling local energy from an explicit probability distribution given by the wavefunction. In this work, we provide a new optimization framework for obtaining properties of quantum many-body ground states using score-based neural networks. Our new framework does not require explicit probability distribution and performs the sampling via Langevin dynamics. Our method is based on the key observation that the local energy is directly related to scores, defined as the gradient of the logarithmic wavefunction. Inspired by the score matching and diffusion Monte Carlo methods, we derive a weighted score matching objective to guide our score-based models to converge correctly to ground states. We first evaluate our approach with experiments on quantum harmonic traps, and results show that it can accurately learn ground states of atomic systems. By implicitly modeling high-dimensional data distributions, our work paves the way toward a more efficient representation of quantum systems.
Quantum Monte Carlo coupled with neural network wavefunctions has shown success in computing ground states of quantum many-body systems. Existing optimization approaches compute the energy by sampling local energy from an explicit probability distribution given by the wavefunction. In this work, we provide a new optimization framework for obtaining properties of quantum many-body ground states using score-based neural networks. Our new framework does not require explicit probability distribution and performs the sampling via Langevin dynamics. Our method is based on the key observation that the local energy is directly related to scores, defined as the gradient of the logarithmic wavefunction. Inspired by the score matching and diffusion Monte Carlo methods, we derive a weighted score matching objective to guide our score-based models to converge correctly to ground states. We first evaluate our approach with experiments on quantum harmonic traps, and results show that it can accurately learn ground states of atomic systems. By implicitly modeling high-dimensional data distributions, our work paves the way toward a more efficient representation of quantum systems.
Neural quantum states (NQS) have gained prominence in variational quantum Monte Carlo methods in approximating ground-state wavefunctions. Despite their success, they face limitations in optimization, scalability, and expressivity in addressing certain problems. In this work, we propose a quantum-neural hybrid framework that combines parametrized quantum circuits with neural networks to model quantum many-body wavefunctions. This approach combines the efficient sampling and optimization capabilities of autoregressive neural networks with the enhanced expressivity provided by quantum circuits. Numerical simulations demonstrate the scalability and accuracy of the hybrid ansatz in spin systems and quantum chemistry problems. Our results reveal that the hybrid method achieves notably lower relative energy compared to stand-alone NQS. These findings underscore the potential of quantum-neural hybrid methods for tackling challenging problems in quantum many-body simulations.
Abstract This paper presents a general neural network framework for solving quantum few-body systems, extending prior methods to handle diverse particle masses, interaction types, and system configurations. Our architecture, which combines an adaptive step size with the Metropolis-Adjusted Langevin Algorithm for Monte Carlo sampling, accurately approximates the ground-state wave functions of systems featuring harmonic confinement, Gaussian two-body interactions, and including three-body forces. In ten-particle systems, it achieves lower relative energy errors (with respect to the reference values) than previous machine-learning methods. Leveraging GPU-accelerated computation, the method scales favorably with system size while maintaining robust convergence, reduced hyperparameter sensitivity, and stable training. Beyond accurate energy estimation, the model captures spatial distributions and correlation structures, offering physical insights about inter-particle structure. By unifying applicability across identical and nonidentical particles, the proposed approach establishes a versatile computational tool for exploring complex few-body quantum systems, with significant implications for advancing computational models in few-body quantum systems.
This paper presents a general neural network framework for solving quantum few-body systems, extending prior methods to handle diverse particle masses, interaction types, and system configurations. Our architecture, which combines an adaptive step size with the Metropolis-Adjusted Langevin Algorithm for Monte Carlo sampling, accurately approximates the ground-state wave functions of systems featuring harmonic confinement, Gaussian two-body interactions, and including three-body forces. In ten-particle systems, it achieves lower relative energy errors (with respect to the reference values) than previous machine-learning methods. Leveraging GPU-accelerated computation, the method scales favorably with system size while maintaining robust convergence, reduced hyperparameter sensitivity, and stable training. Beyond accurate energy estimation, the model captures spatial distributions and correlation structures, offering physical insights about inter-particle structure. By unifying applicability across identical and nonidentical particles, the proposed approach establishes a versatile computational tool for exploring complex few-body quantum systems, with significant implications for advancing computational models in few-body quantum systems.
The potential energy surface (PES) of molecules with respect to their nuclear positions is a primary tool in understanding chemical reactions from first principles. However, obtaining this information is complicated by the fact that sampling a large number of ground states over a high-dimensional PES can require a vast number of state preparations. In this work, we propose using a generative quantum machine learning model to prepare quantum states at arbitrary points on the PES. The model is trained using quantum data consisting of ground-state wavefunctions associated with different classical nuclear coordinates. Our approach uses a classical neural network to convert the nuclear coordinates of a molecule into quantum parameters of a variational quantum circuit. The model is trained using a fidelity loss function to optimize the neural network parameters. We show that gradient evaluation is efficient and numerically demonstrate our method's ability to prepare wavefunctions on the PES of hydrogen chains, water, and beryllium hydride. In all cases, we find that a small number of training points are needed to achieve very high overlap with the groundstates in practice. From a theoretical perspective, we further prove limitations on these protocols by showing that if we were able to learn across an avoided crossing using a small number of samples, then we would be able to violate Grover's lower bound. Additionally, we prove lower bounds on the amount of quantum data needed to learn a locally optimal neural network function using arguments from quantum Fisher information. This work further identifies that quantum chemistry can be an important use case for quantum machine learning.
Neural network quantum states (NQS), incorporating with variational Monte Carlo (VMC) method, are shown to be a promising way to investigate quantum many-body physics. Whereas vanilla VMC methods perform one gradient update per sample, we introduce a novel objective function with proximal optimization (PO) that enables multiple updates via reusing the mismatched samples. Our VMC-PO method keeps the advantage of the previous importance sampling gradient optimization algorithm [L. Yang, {\it et al}, Phys. Rev. Research {\bf 2}, 012039(R)(2020)] that efficiently uses sampled states. PO mitigates the numerical instabilities during network updates, which is similar to stochastic reconfiguration (SR) methods, but achieves an alternative and simpler implement with lower computational complexity. We investigate the performance of our VMC-PO algorithm for ground-state searching with a 1-dimensional transverse-field Ising model and 2-dimensional Heisenberg antiferromagnet on a square lattice, and demonstrate that the reached ground-state energies are comparable to state-of-the-art results.
Score matching has driven major advances in classical generative learning by enabling models to learn from data without evaluating intractable normalization constants, or partition functions. Yet, extending this principle to quantum learning requires rethinking its foundations, as quantum states are described by noncommuting density operators rather than scalar probabilities. The noncommutativity creates fundamental challenges not only in defining quantum scores, but also in developing a training framework with efficient circuit implementations and rigorous theoretical guarantees. In this work, we bridge this gap by establishing a general quantum score-matching framework with end-to-end theoretical guarantees. Applied to Gibbs-state learning, our approach avoids additional thermal-state preparation and achieves information-theoretically optimal sample complexity in the high-temperature regime for Hamiltonians with bounded locality and interaction degree. This positions score matching as a new route to state-of-the-art performance in learning quantum Gibbs states.
Beyond these theoretical results, numerical simulations show that our method remains effective even when gradients are estimated inaccurately under limited measurement budgets. Experiments on IBM quantum hardware further demonstrate that quantum score matching is NISQ-friendly: without any error mitigation or correction, it reduces the relative Hamiltonian-parameter error from 64% to approximately 10%. Together, these results extend score matching into an experimentally realizable paradigm for quantum-state learning.