Accurate solutions for ab initio quantum chemistry are critical for understanding chemistry on the atomic scale. Recently, optimization methods for neural quantum states (NQSs) based on deep neural networks have shown great potential for improving computational accuracy over traditional methods, but they inevitably introduce the challenge of substantial computational costs. To address this challenge, this study proposed expectation–maximization-based optimization of NQS (EMO-NQS), a novel method developed to accelerate optimizations through cost amortization. Specifically, the EMO-NQS method reformulates time-intensive operations as the E-step, amortizing their costs through replays of efficient network-updating operations in the M-step to reduce the overall computational load. Furthermore, an active learning (AL)-based sampling method was introduced to enhance the robustness of EMO-NQS. The performance of EMO-NQS was validated by optimizing the NQS of the equilibrium structures for various molecules. The results demonstrated that EMO-NQS achieved optimization speeds at least twice as fast as those of the vanilla NQS method without compromising accuracy. Furthermore, the consistent acceleration of optimization and the robustness of accuracy facilitated by the AL-based EMO-NQS method were demonstrated by an additional optimization of the construction of the potential energy surface of the N 2 molecule.Accurate solutions for ab initio quantum chemistry are critical for understanding chemistry on the atomic scale. Recently, optimization methods for neural quantum states (NQSs) based on deep neural networks have shown great potential for improving computational accuracy over traditional methods, but they inevitably introduce the challenge of substantial computational costs. To address this challenge, this study proposed expectation–maximization-based optimization of NQS (EMO-NQS), a novel method developed to accelerate optimizations through cost amortization. Specifically, the EMO-NQS method reformulates time-intensive operations as the E-step, amortizing their costs through replays of efficient network-updating operations in the M-step to reduce the overall computational load. Furthermore, an active learning (AL)-based sampling method was introduced to enhance the robustness of EMO-NQS. The performance of EMO-NQS was validated by optimizing the NQS of the equilibrium structures for various molecules. The results demonstrated that EMO-NQS achieved optimization speeds at least twice as fast as those of the vanilla NQS method without compromising accuracy. Furthermore, the consistent acceleration of optimization and the robustness of accuracy facilitated by the AL-based EMO-NQS method were demonstrated by an additional optimization of the construction of the potential energy surface of the N 2 molecule.

Journal of Chemical Theory and ComputationMay 20, 20251 citation

Empowering deep neural quantum states through efficient optimization

Ao Chen, Markus Heyl

Abstract 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 10 6 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.Abstract 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 10 6 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.

arXiv:2302.01941Nature PhysicsJul 1, 202488 citations

Essentially No Energy Barrier Between Independent Fermionic Neural Quantum State Minima

David D. Dai, Marin Soljačić

Neural quantum states (NQS) have proven highly effective in representing quantum many-body wavefunctions, but their loss landscape remains poorly understood and debated. Here, we demonstrate that the NQS loss landscape is more benign and similar to conventional deep learning than previously thought, exhibiting mode connectivity: independently trained NQS are connected by paths in parameter space with essentially no energy barrier. To construct these paths, we develop GeoNEB, a path optimizer integrating efficient stochastic reconfiguration with the nudged elastic band method for constructing minimum energy paths. For the strongly interacting six-electron quantum dot modeled by a $1.6$ M-parameter Psiformer, we find two independent minima with expected energy barrier $\sim10^{-5}$ times smaller than the system's overall energy scale and $\sim10^{-3}$ times smaller than the linear path's barrier. The path respects physical symmetry in addition to achieving low energy, with the angular momentum remaining well quantized throughout. Our work is the first to construct optimized paths between independently trained NQS, and it suggests that the NQS loss landscape may not be as pathological as once feared.Neural quantum states (NQS) have proven highly effective in representing quantum many-body wavefunctions, but their loss landscape remains poorly understood and debated. Here, we demonstrate that the NQS loss landscape is more benign and similar to conventional deep learning than previously thought, exhibiting mode connectivity: independently trained NQS are connected by paths in parameter space with essentially no energy barrier. To construct these paths, we develop GeoNEB, a path optimizer integrating efficient stochastic reconfiguration with the nudged elastic band method for constructing minimum energy paths. For the strongly interacting six-electron quantum dot modeled by a $1.6$M-parameter Psiformer, we find two independent minima with expected energy barrier $\sim10^{-5}$ times smaller than the system's overall energy scale and $\sim10^{-3}$ times smaller than the linear path's barrier. The path respects physical symmetry in addition to achieving low energy, with the angular momentum remaining well quantized throughout. Our work is the first to construct optimized paths between independently trained NQS, and it suggests that the NQS loss landscape may not be as pathological as once feared.

cond-mat.dis-nnCross-listedarXiv:2601.06939Jan 13, 20260 citations

Deep neural networks for quantum state characterization, part 2: reconstruction

Shahnawaz Ahmed, Carlos Sánchez Muñoz, Franco Nori +1 more

Bulletin of the American Physical SocietyMar 16, 20210 citations

Simulations of quantum neural networks

Elizabeth Behrman, L.R. Nash, James E. Steck +2 more

Information SciencesOct 1, 2000123 citations

1.6

M-parameter Psiformer, we find two independent minima with expected energy barrier

\sim10^{-5}

times smaller than the system's overall energy scale and

\sim10^{-3}

times smaller than the linear path's barrier. The path respects physical symmetry in addition to achieving low energy, with the angular momentum remaining well quantized throughout. Our work is the first to construct optimized paths between independently trained NQS, and it suggests that the NQS loss landscape may not be as pathological as once feared.