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Chinese Physics Letters

Solving Quantum Many-Particle Models with Graph Attention Network

Qi-Hang 启航 Yu 于, Zi-Jing 子敬 Lin 林

Published Mar 1, 2024• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/41/3/030202

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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.
Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects.
Deep learning methods have been shown to be effective in representing ground-state wave functions of quantum many-body systems. Existing methods use convolutional neural networks (CNNs) for square lattices due to their image-like structures. For non-square lattices, the existing method uses graph neural networks (GNNs) in which structure information is not precisely captured, thereby requiring additional hand-crafted sublattice encoding. In this work, we propose lattice convolutions in which a set of proposed operations are used to convert non-square lattices into gridlike augmented lattices on which regular convolution can be applied. Based on the proposed lattice convolutions, we design lattice convolutional networks (LCN) that use self-gating and attention mechanisms. Experimental results show that our method achieves performance on par or better than the GNN method on spin 1/2 J1-J2 Heisenberg model over the square, honeycomb, triangular, and kagome lattices while without using hand-crafted encoding. The code will be made publicly available at https://github.com/divelab/AIRS/tree/main/OpenQM/LCN.
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.
We introduce Quantum Graph Attention Networks (QGATs) as trainable quantum encoders for inductive learning on graphs, extending the Quantum Graph Neural Networks (QGNN) framework. QGATs leverage parameterized quantum circuits to encode node features and neighborhood structures, with quantum attention mechanisms modulating the contribution of each neighbor via dynamically learned unitaries. This allows for expressive, locality-aware quantum representations that can generalize across unseen graph instances. We evaluate our approach on the QM9 dataset, targeting the prediction of various chemical properties. Our experiments compare classical and quantum graph neural networks-with and without attention layers-demonstrating that attention consistently improves performance in both paradigms. Notably, we observe that quantum attention yields increasing benefits as graph size grows, with QGATs significantly outperforming their non-attentive quantum counterparts on larger molecular graphs. Furthermore, for smaller graphs, QGATs achieve predictive accuracy comparable to classical GAT models, highlighting their viability as expressive quantum encoders. These results show the potential of quantum attention mechanisms to enhance the inductive capacity of QGNN in chemistry and beyond.
The quantum many-body problem is ultimately a curse of dimensionality: the state of a system with many particles is determined by a function with many dimensions, which rapidly becomes difficult to efficiently store, evaluate and manipulate numerically. On the other hand, modern machine learning models like deep neural networks can express highly correlated functions in extremely large-dimensional spaces, including those describing quantum mechanical problems. We show that if one represents wavefunctions as a stochastically generated set of sample points, the problem of finding ground states can be reduced to one where the most technically challenging step is that of performing regression-a standard supervised learning task. In the stochastic representation the (anti)symmetric property of fermionic/bosonic wavefunction can be used for data augmentation and learned rather than explicitly enforced. We further demonstrate that propagation of an ansatz towards the ground state can then be performed in a more robust and computationally scalable fashion than traditional variational approaches allow.
The attention mechanism has transformed artificial intelligence research by its ability to learn relations between objects. In this work, we explore how a many-body wavefunction ansatz constructed from a large-parameter self-attention neural network can be used to solve the interacting electron problem in solids. By a systematic neural-network variational Monte Carlo study on a moiré quantum material, we demonstrate that the self-attention ansatz provides an accurate and efficient solution without human bias. Moreover, our numerical study finds that the required number of variational parameters scales roughly as N2N^2N2 with the number of electrons, which opens a path towards efficient large-scale simulations.