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The European Physical Journal Plus

Calculation of entanglement entropy of transverse-field ising model from neural network quantum state based on a restricted Boltzmann machine

Zhekai Yin, Zixuan Li, Fei Fang Chung, Kok Wee Song, C. K. Ong

Published Oct 3, 2025• 0 citations
Publisher
DOI
10.1140/epjp/s13360-025-06855-3

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Recently, quantum-state representation using artificial neural networks has started to be recognized as a powerful tool. However, due to the black-box nature of machine learning, it is difficult to analyze what machine learns or why it is powerful. Here, by applying one of the simplest neural networks, the restricted Boltzmann machine (RBM), to the ground-state representation of the one-dimensional (1D) transverse-field Ising (TFI) model, we make an attempt to directly analyze the optimized network parameters. In the RBM optimization, a zero-temperature quantum state is mapped onto a finite-temperature classical state of the extended Ising spins that constitute the RBM. We find that the quantum phase transition from the ordered phase to the disordered phase in the 1D TFI model by increasing the transverse field is clearly reflected in the behaviors of the optimized RBM parameters and hence in the finite-temperature phase diagram of the classical RBM Ising system. The present finding of a correspondence between the neural-network parameters and quantum phases suggests that a careful investigation of the neural-network parameters may provide a new route to extracting nontrivial physical insights from the neural-network wave functions.
We propose a Restricted Boltzmann Machine (RBM) neural network using a quantum thermodynamics formalism and the maximization of entropy as the cost function for the optimization problem. We verify the possibility of using an entropy maximization approach for modeling physical systems by applying it to the Inverse Ising Problem, which describes a system of interacting spins. The temperature-dependent behaviors of physical quantities, such as magnetization and Gibbs energy, were calculated from the Ising Hamiltonian whose coefficients were obtained through entropy maximization and were found to be in good agreement to literature results. These results suggest that RBM neural networks using the principle of maximum entropy can be applied to modeling physical systems that can be described by discrete states, including fundamental quantum physics such as topological systems, and biological systems such as the correlated spiking of neurons.
Abstract Along the way initiated by Carleo and Troyer [G. Carleo and M. Troyer, Science 355 (2017) 602], we construct the neural-network quantum state of transverse-field Ising model (TFIM) by an unsupervised machine learning method. Such a wave function is a map from the spin-configuration space to the complex number field determined by an array of network parameters. To get the ground state of the system, values of the network parameters are calculated by a Stochastic Reconfiguration (SR) method. We provide for this SR method an understanding from action principle and information geometry aspects. With this quantum state, we calculate key observables of the system, the energy, correlation function, correlation length, magnetic moment, and susceptibility. As innovations, we provide a high efficiency method and use it to calculate entanglement entropy (EE) of the system and get results consistent with previous work very well .