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arXiv:2206.07370 · Society for Industrial and Applied Mathematics eBooks

Lattice Convolutional Networks for Learning Ground States of Quantum Many-Body Systems

Cong Fu, Xuan Zhang, Huixin Zhang, Hongyi Ling, Shenglong Xu, Shuiwang Ji

Published Jan 1, 2024• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1137/1.9781611978032.57

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Abstract Quantum neural networks (QNNs) represent a pioneering intersection of quantum computing and deep learning. In this study, we unveil a fundamental convolution property inherent to QNNs, stemming from the natural parallelism of quantum gate operations on quantum states. Notably, QNNs are capable of performing a convolutional layer using a single quantum gate, whereas classical methods require 2n2^n2n basic operations. This essential property has been largely overlooked in the design of existing quantum convolutional neural networks (QCNNs), limiting their ability to capture key structural features of classical CNNs, including local connectivity, parameter sharing, and multi-channel, multi-layer architectures. To address these limitations, we propose novel QCNN architectures that explicitly harness the convolutional nature of QNNs. We validate the effectiveness of these architectures through extensive numerical experiments focused on multiclass image classification. Our findings provide deep insights into the realization of convolutional mechanisms within QNNs, marking a substantial advancement in the development of QCNNs and broadening their potential for efficient data processing.
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In the past decades, methods to solve the ground state given a quantum many-body Hamiltonian have been well established. In this work, we consider an inverse problem and demonstrate that convolutional neural network (CNN) can predict the physical parameters of interacting Hamiltonians, such as coupling strengths and magnetic fields, providing the quantum many-body wave-functions as the ground states. We propose QubismNet that consists of two main parts: the Qubism map that visualizes the ground states (or the purified reduced density matrices) as images, and a CNN that maps the images to the target physical parameters. QubismNet exhibits impressive powers of learning and generalization on several quantum spin models. While the training samples are restricted to the states from certain ranges of the parameters, QubismNet can accurately predict the parameters of the states beyond such training regions. For instance, our results show that QubismNet can predict the magnetic fields near the critical point by learning from the states away from the critical vicinity. Our work provides a data-driven way to infer the Hamiltonians that give the designed ground states, and therefore would benefit the existing and future generations of quantum technologies such as Hamiltonian-based quantum simulations.