Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.20444

Learning the Renyi entropy of multiple disjoint intervals in transverse-field quantum Ising models with restricted Boltzmann machine

Han-Qing Shi, Hai-Qing Zhang

Published Dec 29, 2024• 0 citations
arXivHTMLPDF

Abstract

Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: one is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one-dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.

DOI
10.48550/arxiv.2412.20444

Related papers

Learning the Renyi entropy of multiple disjoint intervals in transverse-field quantum Ising models with a restricted Boltzmann machine

Han-Qing Shi, Hai-Qing Zhang

Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: One is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three, and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.

Physical review. E•Nov 17, 2025•0 citations

A universal approach to Renyi entropy of multiple disjoint intervals

Han-Qing Shi, Hai-Qing Zhang

We propose a universal approach to compute the Renyi entropy with generically multiple disjoint intervals from the swapping operations. The proposal is grounded in our observation that there is a striking similarity between the replica trick utilized in quantum field theory and the process of swapping operations. Consequently, the Renyi entropy of multiple disjoint intervals can be obtained by evaluating the expectation values of the swapping operator. As an application, we use our proposal to study the Renyi entropy of a one-dimensional transverse-field Ising model with two, three and four disjoint intervals. In particular, as the system is at the critical point, our results of the Renyi entropy are consistent with the analytical results from the conformal field theory. Remarkably, our approach can go beyond the critical regime where the analytical methods from conformal field theory can not apply.

arXiv:2411.18353•Physics Letters B•Mar 28, 2026•0 citations

Learning Rényi entropy in neural network quantum states

Han-Qing Shi, Hai-Qing Zhang

Abstract We study the Rényi entropy in a one-dimensional transverse-field quantum Ising model by virtue of the machine learning methods with neural networks. The Rényi entropy is investigated both in the static ground state and in the dynamical state after a linear quench, by employing a swapping operator acting on the states prepared from the neural networks. In the ground state, the behavior of Rényi entropy can unveil the critical point from paramagnetic to ferromagnetic phase. At the critical point, the relation between the Rényi entropy and the subsystem size from the machine learning method satisfies the predictions from conformal field theory. In the quenching dynamics, coherent oscillations of the Rényi entropy are observed after the end of the quench. These oscillations have universal frequencies stem from the superpositions of the excited states. More importantly, an asymptotic scaling of the reduced Rényi entropy is observed, implying a novel length scale away from the critical point.

Physica Scripta•Oct 21, 2026•0 citations

Measuring Renyi Entropy in Neural Network Quantum States

Han-Qing Shi, Hai-Qing Zhang

We compute the Renyi entropy in a one-dimensional transverse-field quantum Ising model by employing a swapping operator acting on the states which are prepared from the neural network methods. In the static ground state, Renyi entropy can uncover the critical point of the quantum phase transition from paramagnetic to ferromagnetic. At the critical point, the relation between the Renyi entropy and the subsystem size satisfies the predictions from conformal field theory. In the dynamical case, we find coherent oscillations of the Renyi entropy after the end of the linear quench. These oscillations have universal frequencies which may come from the superpositions of excited states. The asymptotic form of the Renyi entropy implies a new length scale away from the critical point. This length scale is also verified by the overlap of the reduced Renyi entropy against the dimensionless subsystem size.

arXiv:2308.05513•Aug 10, 2023•0 citations

Quantum boltzmann machine :emergence & applications

Miguel Francisco Aldana Páez

Lately, there has been an extensive state-of-the-art research in Machine Learning methods, due to their important features such as universality in approximations and dimensional reduction. In this way, the present work aims at exploiting these properties of Machine Learning on physical many-body problems, in which a restricted Boltzmann Machine (RBM) is trained with quantum Monte Carlo data to best represent the ground state of an N spin system under the transverse field Ising model Hamiltonian in one dimension. During the path coursed along the development of this project, we first review classical well-implemented methods such as the single-layer perceptron, multi-layer perceptron for the XOR problem with back-propagation algorithm, discrete Hopfield network for pattern reconstruction, continuous Hopfield network with Simulated Annealing for solving the traveling salesman problem with 10 cities and the Boltzmann machine for feature extraction. Then, we review a few aspects of what is called a Quantum Boltzmann Machine and the basics of the transverse field Ising model. Finally, we train an RBM consisting of N visible neurons, hidden density neurons alpha=M/N=2 and 1-step contrastive divergence (CD_1), by minimizing the Kullbach-Liebler divergence of the RBM and the training data set coming from the algorithmic variant of quantum Monte Carlo, known as discrete Path-Integral Monte Carlo. For this last step, we use Trotter-Suzuki decomposition with m Trotter slices and the Wolff-s cluster algorithm. After training the RBM, we calculate magnetic observables based on the RBM-s wave-function such as longitudinal and transverse magnetization, for different values of transverse field and we do an estimation of the critical field at which the transverse Ising model system suffers a quantum phase transition from ferromagnet to paramagnet

•Jan 1, 2018•0 citations

Investigating Network Parameters in Neural-Network Quantum States

Yusuke Nomura

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.

arXiv:2202.01704•Journal of the Physical Society of Japan•Apr 18, 2022•5 citations
Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: one is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one-dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.
Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: One is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three, and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.
We propose a universal approach to compute the Renyi entropy with generically multiple disjoint intervals from the swapping operations. The proposal is grounded in our observation that there is a striking similarity between the replica trick utilized in quantum field theory and the process of swapping operations. Consequently, the Renyi entropy of multiple disjoint intervals can be obtained by evaluating the expectation values of the swapping operator. As an application, we use our proposal to study the Renyi entropy of a one-dimensional transverse-field Ising model with two, three and four disjoint intervals. In particular, as the system is at the critical point, our results of the Renyi entropy are consistent with the analytical results from the conformal field theory. Remarkably, our approach can go beyond the critical regime where the analytical methods from conformal field theory can not apply.
Abstract We study the Rényi entropy in a one-dimensional transverse-field quantum Ising model by virtue of the machine learning methods with neural networks. The Rényi entropy is investigated both in the static ground state and in the dynamical state after a linear quench, by employing a swapping operator acting on the states prepared from the neural networks. In the ground state, the behavior of Rényi entropy can unveil the critical point from paramagnetic to ferromagnetic phase. At the critical point, the relation between the Rényi entropy and the subsystem size from the machine learning method satisfies the predictions from conformal field theory. In the quenching dynamics, coherent oscillations of the Rényi entropy are observed after the end of the quench. These oscillations have universal frequencies stem from the superpositions of the excited states. More importantly, an asymptotic scaling of the reduced Rényi entropy is observed, implying a novel length scale away from the critical point.
We compute the Renyi entropy in a one-dimensional transverse-field quantum Ising model by employing a swapping operator acting on the states which are prepared from the neural network methods. In the static ground state, Renyi entropy can uncover the critical point of the quantum phase transition from paramagnetic to ferromagnetic. At the critical point, the relation between the Renyi entropy and the subsystem size satisfies the predictions from conformal field theory. In the dynamical case, we find coherent oscillations of the Renyi entropy after the end of the linear quench. These oscillations have universal frequencies which may come from the superpositions of excited states. The asymptotic form of the Renyi entropy implies a new length scale away from the critical point. This length scale is also verified by the overlap of the reduced Renyi entropy against the dimensionless subsystem size.
Lately, there has been an extensive state-of-the-art research in Machine Learning methods, due to their important features such as universality in approximations and dimensional reduction. In this way, the present work aims at exploiting these properties of Machine Learning on physical many-body problems, in which a restricted Boltzmann Machine (RBM) is trained with quantum Monte Carlo data to best represent the ground state of an N spin system under the transverse field Ising model Hamiltonian in one dimension. During the path coursed along the development of this project, we first review classical well-implemented methods such as the single-layer perceptron, multi-layer perceptron for the XOR problem with back-propagation algorithm, discrete Hopfield network for pattern reconstruction, continuous Hopfield network with Simulated Annealing for solving the traveling salesman problem with 10 cities and the Boltzmann machine for feature extraction. Then, we review a few aspects of what is called a Quantum Boltzmann Machine and the basics of the transverse field Ising model. Finally, we train an RBM consisting of N visible neurons, hidden density neurons alpha=M/N=2 and 1-step contrastive divergence (CD_1), by minimizing the Kullbach-Liebler divergence of the RBM and the training data set coming from the algorithmic variant of quantum Monte Carlo, known as discrete Path-Integral Monte Carlo. For this last step, we use Trotter-Suzuki decomposition with m Trotter slices and the Wolff-s cluster algorithm. After training the RBM, we calculate magnetic observables based on the RBM-s wave-function such as longitudinal and transverse magnetization, for different values of transverse field and we do an estimation of the critical field at which the transverse Ising model system suffers a quantum phase transition from ferromagnet to paramagnet
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.