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arXiv:2112.00723 · Machine Learning Science and Technology

Infinite neural network quantum states: entanglement and training dynamics

Di Luo, James Halverson

Published Jun 1, 2023• 7 citations
arXivHTMLPDF

Abstract

Abstract We study infinite limits of neural network quantum states ( ∞ -NNQS), which exhibit representation power through ensemble statistics, and also tractable gradient descent dynamics. Ensemble averages of entanglement entropies are expressed in terms of neural network correlators, and architectures that exhibit volume-law entanglement are presented. The analytic calculations of entanglement entropy bound are tractable because the ensemble statistics are simplified in the Gaussian process limit. A general framework is developed for studying the gradient descent dynamics of neural network quantum states (NNQS), using a quantum state neural tangent kernel (QS-NTK). For ∞ -NNQS the training dynamics is simplified, since the QS-NTK becomes deterministic and constant. An analytic solution is derived for quantum state supervised learning, which allows an ∞ -NNQS to recover any target wavefunction. Numerical experiments on finite and infinite NNQS in the transverse field Ising model and Fermi Hubbard model demonstrate excellent agreement with theory. ∞ -NNQS opens up new opportunities for studying entanglement and training dynamics in other physics applications, such as in finding ground states.

DOI
10.1088/2632-2153/ace02f

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Abstract We study infinite limits of neural network quantum states ( ∞ -NNQS), which exhibit representation power through ensemble statistics, and also tractable gradient descent dynamics. Ensemble averages of entanglement entropies are expressed in terms of neural network correlators, and architectures that exhibit volume-law entanglement are presented. The analytic calculations of entanglement entropy bound are tractable because the ensemble statistics are simplified in the Gaussian process limit. A general framework is developed for studying the gradient descent dynamics of neural network quantum states (NNQS), using a quantum state neural tangent kernel (QS-NTK). For ∞ -NNQS the training dynamics is simplified, since the QS-NTK becomes deterministic and constant. An analytic solution is derived for quantum state supervised learning, which allows an ∞ -NNQS to recover any target wavefunction. Numerical experiments on finite and infinite NNQS in the transverse field Ising model and Fermi Hubbard model demonstrate excellent agreement with theory. ∞ -NNQS opens up new opportunities for studying entanglement and training dynamics in other physics applications, such as in finding ground states.
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Quantum variational algorithms have been one of major applications of quantum computing with current quantum devices. There are recent attempts to establish the foundation for these algorithms. A possible approach is to characterize the training dynamics with quantum neural tangent kernel. In this work, we construct the kernel for two models, Quantun Ensemble and Quantum Neural Network, and show the convergence of these models in the limit of infinitely many qubits. We also show applications of the kernel limit in regression tasks.
Abstract Deep neural networks have demonstrated remarkable efficacy in extracting meaningful representations from complex datasets. This has propelled representation learning as a compelling area of research across diverse fields. One interesting open question is how beneficial representation learning can be for quantum many-body physics, with its notoriously high-dimensional state space. In this work, we showcase the capacity of a neural network that was trained on a subset of physical observables of a many-body system to partially acquire an implicit representation of the wave function. We illustrate this by demonstrating the effectiveness of reusing the representation learned by the neural network to enhance the learning process of another quantity derived from the quantum state. In particular, we focus on how the pre-trained neural network can enhance the learning of entanglement entropy. This is of particular interest as directly measuring the entanglement in a many-body system is very challenging, while a subset of physical observables can be easily measured in experiments. We show the pre-trained neural network learns the dynamics of entropy with fewer resources and higher precision in comparison with direct training on the entanglement entropy.
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Quantum many-body computation is a central challenge across multiple domains of scientific research, and Monte Carlo method has emerged as one of the most effective numerical techniques for investigating large-scale many-body systems. In recent years, the integration of Monte Carlo methods with machine learning has led to the development of neural quantum state approach. Quantum neural networks (QNNs) offer superior expressiveness compared to classical neural networks. In this paper, we propose a novel Quantum Neural Network Quantum State (QNNQS) method, based on the variational Monte Carlo framework. This approach leverages a quantum neural network to represent the many-body quantum state, and optimizes it iteratively through gradient descent to determine the ground-state energy. To assess the performance of this method, we conduct simulations of the one-dimensional transverse-field Ising model, investigating the influence of quantum circuit architecture on circuit expressibility, entangling capability, and computational accuracy. The results demonstrate that the QNNQS method achieves comparable simulation accuracy while utilizing significantly fewer parameters and requiring less training resources than classical network quantum state methods.