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arXiv:2609.37733

Foundation Neural-Network Quantum States for Molecular Potential Energy Surfaces in Second Quantization

Lizhong Fu, Jianan Wei, Wenguan Wang, Honghui Shang

physics.chem-phcs.LGquant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

Second-quantized neural-network quantum states have achieved accurate molecular energies, but extending them across molecular geometries requires a shared representation of the geometry-dependent wavefunction coefficients. We introduce geometry-conditioned foundation neural-network quantum states for molecular electronic structure in second quantization. A single autoregressive model learns a family of ground states from sparse anchor geometries and provides wavefunctions at untrained geometries without further optimization. Orbital alignment matches orbital identities and transports their phases, establishing an aligned orbital basis across geometries. Frozen energies reach chemical accuracy at every untrained query geometry for N$_2$, CO, and H$_4$. On additional molecular paths, the energy-trained wavefunctions yield dipoles, quadrupoles, and natural occupations without property labels. Across three paired N$_2$ training seeds, orbital alignment lowers the mean absolute energy error over all untrained query geometries from 34-37 mHa to 0.049-0.085 mHa. At approximately 1 mHa mean absolute error, frozen evaluation reduces the per-geometry cost by $986\times$ relative to independent optimization, yielding an estimated $25.8\times$ end-to-end GPU-cost reduction on a 161-point N$_2$ grid.

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23 pages, 7 figures

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Second-quantized neural-network quantum states have achieved accurate molecular energies, but extending them across molecular geometries requires a shared representation of the geometry-dependent wavefunction coefficients. We introduce geometry-conditioned foundation neural-network quantum states for molecular electronic structure in second quantization. A single autoregressive model learns a family of ground states from sparse anchor geometries and provides wavefunctions at untrained geometries without further optimization. Orbital alignment matches orbital identities and transports their phases, establishing an aligned orbital basis across geometries. Frozen energies reach chemical accuracy at every untrained query geometry for N2_22​, CO, and H4_44​. On additional molecular paths, the energy-trained wavefunctions yield dipoles, quadrupoles, and natural occupations without property labels. Across three paired N2_22​ training seeds, orbital alignment lowers the mean absolute energy error over all untrained query geometries from 34-37 mHa to 0.049-0.085 mHa. At approximately 1 mHa mean absolute error, frozen evaluation reduces the per-geometry cost by 986×986\times986× relative to independent optimization, yielding an estimated 25.8×25.8\times25.8× end-to-end GPU-cost reduction on a 161-point N2_22​ grid.
Variational quantum eigensolver ansätze hold considerable promise for ground-state energy calculations on near-term quantum hardware, yet most promising ansatz designs currently strongly depend on how well the molecular orbital basis captures the electronic correlation of the system. Computing optimized orbital coefficients via classical routines is computationally expensive and must be performed independently for each molecular geometry -- a bottleneck that limits scalability across chemical space. We present a graph neural network framework that predicts optimized orbital coefficients directly from molecular geometry and pair-wise bonding structure. Trained on hydrogenic systems of modest size (H4H_4H4​ and H6H_6H6​) across tens of thousands of geometries, our model transfers to larger, unseen systems (H8H_8H8​, H10H_{10}H10​ and H12H_{12}H12​) without retraining -- demonstrating strong out-of-distribution generalization with respect to system size. When evaluating on structured and random configurations, and comparing against energies obtained with full classical optimization, our model reaches mean absolute energy errors O(102)\mathcal{O}(10^2)O(102) and O(10)\mathcal{O}(10)O(10) milli-Hartrees, respectively. Beyond energy estimation, the predicted orbitals serve as high-quality warm-start initializations that substantially reduce optimizer iterations to ground-state energy convergence. These results establish graph neural networks as an effective and scalable strategy for accelerating orbital optimization in hybrid quantum-classical workflows, directly reducing the classical pre-processing overhead that currently limits the practical deployment of variational quantum eigensolver on near-term quantum hardware.
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