Learning ground states of quantum Hamiltonians with graph networks

Abstract

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.

Artificial intelligence in recommender systems

“…Later on, convolutional neural networks (CNNs) [Liang et al 2018;Choo et al 2019;Szabó and Castelnovo 2020] are applied to 2D square lattices and are found to represent highly entangled systems effectively. However, CNN cannot be naturally used on non-grid lattices or even random graphs, which necessitated the exploration of graph neural networks (GNNs) [Yang et al 2020;Kochkov et al 2021a] for dealing with arbitrary geometric lattices. Moreover, autoregressive and recurrent neural networks (RNNs) are applied to represent quantum states, enabling direct sampling of spin configurations [Sharir et al 2020;Hibat-Allah et al 2020;].…”

“…They use the cosine function as the activation function for predicting the sign, which is suitable for capturing the oscillating features of the input spins. Kochkov et al [2021a] predicts log amplitude and phase of wave functions separately and shows that predicting phase directly enables effective generalization of the learned sign structure. Szabó and Castelnovo [2020] models amplitude and sign structure using two real-valued neural networks.…”

ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphs

“…In machine learning, structured objects such as molecules [14,50], proteins [9,16,33,37,28], materials [51], and quantum systems [31] are usually modeled as graphs. Original modeling shows basic connections between units and the resulted data type is known as 2D graphs.…”

Learning a compass spin model with neural network quantum states

“…The J 1 -J 2 model has also been investigated in Refs. [12][13][14][15][16] using the neural network ansatz. In a broader context, recent work has revealed a remarkable connection between the neural-network and the tensornetwork representation of quantum many-body states despite their differences in appearance and origin [17][18][19].…”