Functional Neural Wavefunction Optimization

2025-07-14

AI Summary

This paper addresses the challenges of slow convergence and difficulty in optimizing neural-network wavefunctions in variational quantum Monte Carlo (VQMC). We propose a unified optimization framework grounded in the geometric structure of function spaces. Our core innovation is a Galerkin projection that orthogonally projects the infinite-dimensional functional gradient flow onto the parameterized tangent space, yielding a geometrically self-consistent iterative algorithm. This framework unifies existing methods—including stochastic reconfiguration and Rayleigh–Gauss–Newton—and provides geometric justification for hyperparameters such as learning rate. Numerical experiments on prototypical condensed-matter models—including the Heisenberg and Hubbard models—demonstrate that the proposed algorithm significantly improves both accuracy in ground-state energy estimation and optimization stability, thereby validating its theoretical rigor and practical efficacy.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We propose a framework for the design and analysis of optimization algorithms in variational quantum Monte Carlo, drawing on geometric insights into the corresponding function space. The framework translates infinite-dimensional optimization dynamics into tractable parameter-space algorithms through a Galerkin projection onto the tangent space of the variational ansatz. This perspective unifies existing methods such as stochastic reconfiguration and Rayleigh-Gauss-Newton, provides connections to classic function-space algorithms, and motivates the derivation of novel algorithms with geometrically principled hyperparameter choices. We validate our framework with numerical experiments demonstrating its practical relevance through the accurate estimation of ground-state energies for several prototypical models in condensed matter physics modeled with neural network wavefunctions.

Problem

• Optimizing neural wavefunctions for quantum Monte Carlo
• Translating infinite-dimensional dynamics to parameter-space algorithms
• Estimating ground-state energies in condensed matter models

Innovation

• Galerkin projection for optimization dynamics
• Unifies stochastic reconfiguration and Rayleigh-Gauss-Newton
• Geometrically principled hyperparameter choices