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arXiv:2011.12453

Scalable variational Monte Carlo with graph neural ansatz

Li Yang, Wen-Jun Hu, Li, Li

Published Nov 25, 2020• 9 citations
arXivHTMLPDF

Abstract

Deep neural networks have been shown as a potentially powerful ansatz in variational Monte Carlo for solving quantum many-body problems. We propose two improvements in this direction. The first is graph neural ansatz (GNA), which is a variational wavefunction universal to arbitrary geometry. GNA results in accurate ground-state energies on 2D Kagome lattices, triangular lattices, and randomly connected graphs. Secondly, we design a distributed workflow on multiple accelerators to scale up the computation. We compute Kagome lattices with sizes up to 432 sites on 128 TPU cores. The parameter sharing nature of the GNA also leads to transferability across different system sizes and geometries.

DOI
10.48550/arxiv.2011.12453

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Deep neural networks have been shown as a potentially powerful ansatz in variational Monte Carlo for solving quantum many-body problems. We propose two improvements in this direction. The first is graph neural ansatz (GNA), which is a variational wavefunction universal to arbitrary geometry. GNA results in accurate ground-state energies on 2D Kagome lattices, triangular lattices, and randomly connected graphs. Secondly, we design a distributed workflow on multiple accelerators to scale up the computation. We compute Kagome lattices with sizes up to 432 sites on 128 TPU cores. The parameter sharing nature of the GNA also leads to transferability across different system sizes and geometries.
Deep learning methods have been shown to be effective in representing ground-state wavefunctions of quantum many-body systems, however the existing approaches cannot be easily used for non-square like or large systems. Here, we propose a variational ansatz based on the graph attention network (GAT) which learns distributed latent representations and can be used on non-square lattices. The GAT-based ansatz has a computational complexity that grows linearly with the system size and can be extended to large systems naturally. Numerical results show that our method achieves the state-of-the-art results on spin-1/2 J 1–J 2 Heisenberg models over the square, honeycomb, triangular, and kagome lattices with different interaction strengths and lattice sizes (up to 24 × 24 for square lattice). The method also provides excellent results for the ground states of transverse field Ising models on square lattices. The GAT-based techniques are efficient and versatile and hold promise for studying large quantum many-body systems with exponentially sized objects.
Abstract Deep-learning-based variational Monte Carlo has emerged as a highly accurate method for solving the many-electron Schrödinger equation. Despite favorable scaling with the number of electrons, O(nel4){\mathcal{O}}({{n}_{{\rm{el}}}}^{4})O(nel​4) O ( n el 4 ) , the practical value of deep-learning-based variational Monte Carlo is limited by the high cost of optimizing the neural network weights for every system studied. Recent research has proposed optimizing a single neural network across multiple systems, reducing the cost per system. Here we extend this approach to solids, which require numerous calculations across different geometries, boundary conditions and supercell sizes. We demonstrate that optimization of a single ansatz across these variations significantly reduces optimization steps. Furthermore, we successfully transfer a network trained on 2 × 2 × 2 supercells of LiH, to 3 × 3 × 3 supercells, reducing the number of optimization steps required to simulate the large system by a factor of 50 compared with previous work.
Neural-network variational Monte Carlo (NNVMC) has emerged as a powerful tool for solving quantum many-body problems, yet systematic pathways for improving its accuracy remain largely heuristic. Here, we introduce a physically motivated basis transformation for NNVMC that enhances variational expressivity without increasing the complexity of the neural-network ansatz itself. By formulating the many-body wave function in a Gaussian basis, we introduce a single learnable locality parameter, ααα, that reshapes the target ground state into a more learnable representation. This approach introduces minimal computational overhead and can be readily combined with existing neural-network architectures. Using the three-dimensional homogeneous electron gas as a benchmark, we show that the optimized basis transformation consistently lowers the variational energy for both FermiNet and message-passing neural-network architectures. Notably, for the latter, it enables a more precise determination of the Fermi liquid to Wigner crystal phase transition. More broadly, our results highlight basis transformation as a new route to improving NNVMC in continuous space, showing that accuracy can be enhanced not only by refining the ansatz but also by making the target ground state easier to represent.
Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles.Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms.The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy.Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics.Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems.The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules.Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry.This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.
Representing a target quantum state by a compact, efficient variational wave-function is an important approach to the quantum many-body problem. In this approach, the main challenges include the design of a suitable variational ansatz and optimization of its parameters. In this work, we address both of these challenges. First, we define the variational class of Computational Graph States (CGS) which gives a uniform framework for describing all computable variational ansatz. Secondly, we develop a novel optimization scheme, supervised wave-function optimization (SWO), which systematically improves the optimized wave-function by drawing on ideas from supervised learning. While SWO can be used independently of CGS, utilizing them together provides a flexible framework for the rapid design, prototyping and optimization of variational wave-functions. We demonstrate CGS and SWO by optimizing for the ground state wave-function of 1D and 2D Heisenberg models on nine different variational architectures including architectures not previously used to represent quantum many-body wave-functions and find they are energetically competitive to other approaches. One interesting application of this architectural exploration is that we show that fully convolution neural network wave-functions can be optimized for one system size and, using identical parameters, produce accurate energies for a range of system sizes. We expect these methods to increase the rate of discovery of novel variational ansatz and bring further insights to the quantum many body problem.
Quantum many-body problems are some of the most challenging problems in science and are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, coupled with the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. However, the run-time of this approach scales quadratically with the number of simulated particles, constraining the practically usable NN to - in machine learning terms - minuscule sizes (<10M parameters). Considering the many breakthroughs brought by extreme NN in the +1B parameters scale to other domains, lifting this constraint could significantly expand the set of quantum systems we can accurately simulate on classical computers, both in size and complexity. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm - the source of the quadratic scaling. In our preliminary experiments, we demonstrate VQ-NQS ability to reproduce the ground state of the 2D Heisenberg model across various system sizes, while reporting a significant reduction of about ×10{\times}10×10 in the number of FLOPs in the local-energy calculation.