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arXiv:2411.02257

Neural network-based nodal structures optimization for interacting fermionic systems

Freitas, William, B De Abreu, S. A. Vitiello

Published Nov 4, 2024• 0 citations
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Abstract

Simulating strongly correlated fermionic systems remains a fundamental challenge in quantum physics, largely due to the sign problem in quantum Monte Carlo (QMC) methods. We present a neural network-based variational Monte Carlo (NN-VMC) approach, leveraging a flexible neural network ansatz to represent the many-body wavefunction. Focusing on quantum dots with up to 30 electrons, we demonstrate that NN-VMC significantly reduces variational bias and achieves ground-state energies surpassing those of fixed-node diffusion Monte Carlo (DMC). A key feature is that the neural network adaptively learns and optimizes nodal structures during energy minimization. We provide qualitative insights into the nodal structure of fermionic wavefunctions by comparing the nodal structures generated by NN-VMC with those obtained from traditional trial functions. Additionally, we reveal spin-resolved radial distributions and electron density profiles, highlighting the versatility and accuracy of NN-VMC. This work underscores the potential of machine learning to advance quantum simulations and deepen our understanding of strongly correlated systems.

DOI
10.48550/arxiv.2411.02257

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The central challenge in materials science and quantum chemistry is solving the electronic Schrodinger equation, complicated by the curse of dimensionality as system size grows. Neural network-based variational Monte Carlo (NN-VMC) offers a promising path forward, achieving unprecedented accuracy at far lower cost than traditional high-level methods. However, the flexibility of neural network wavefunctions introduces a bottleneck: their optimization is high-dimensional, stochastic, and non-convex. This Perspective reviews the evolution of optimization methods in NN-VMC, from stochastic reconfiguration to approximate second-order algorithms and geometric insights. We highlight key challenges currently limiting scalability and efficiency, and outline future opportunities to advance the field. With continued progress in optimization, neural network techniques, and computer architectures, NN-VMC can tackle larger and more complex quantum systems and move from trailing experiments to guiding them.
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We introduce an efficient approach to implement neural network quantum states (NNQS) as trial wave functions in auxiliary-field quantum Monte Carlo (AFQMC). NNQS are a recently developed class of variational ansätze capable of flexibly representing many-body wave functions, though they often incur a high computational cost during optimization. AFQMC, on the other hand, is a powerful stochastic projector approach for ground-state calculations, but it normally requires an approximate constraint via a trial wave function or trial density matrix, whose quality affects the accuracy. Recently, it has been shown (Xiao et al., arXiv2505.18519) that a broad class of highly correlated wave functions can be integrated into AFQMC through stochastic sampling techniques. In this work, we apply this approach and present a direct integration of NNQS with AFQMC, allowing NNQS to serve as high-quality trial wave functions for AFQMC with manageable computational cost. We test the NNQS-AFQMC method on the challenging nitrogen molecule (N 2 ) at stretched geometries. Our results demonstrate that AFQMC with an NNQS trial wave function can attain near-exact total energies, highlighting the potential of AFQMC with NNQS to overcome longstanding challenges in strongly correlated electronic structure calculations. We also outline future research directions for improving this promising methodology.