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arXiv:2408.14306 · Physical Review A

Delta-learning approach combined with the cluster Gutzwiller approximation for strongly correlated bosonic systems

Zhi Lin, Sheng Yue, T. Y. Wang

Published Jan 28, 2026• 0 citations
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Abstract

The cluster Gutzwiller method is widely used to study the strongly correlated bosonic systems, owing to its ability to provide a more precise description of quantum fluctuations. However, its utility is limited by the exponential increase in computational complexity as the cluster size grows. To overcome this limitation, we propose an artificial intelligence-based method known as $Δ$-Learning. This approach constructs a predictive model by learning the discrepancies between lower-precision (small cluster sizes) and high-precision (large cluster sizes) implementations of the cluster Gutzwiller method, requiring only a small number of training samples. Using this predictive model, we can effectively forecast the outcomes of high-precision methods with high accuracy. Applied to various Bose-Hubbard models, the $Δ$-Learning method effectively predicts phase diagrams while significantly reducing the computational resources and time. Furthermore, we have compared the predictive accuracy of $Δ$-Learning with other direct learning methods and found that $Δ$-Learning exhibits superior performance in scenarios with limited training data. Therefore, when combined with the cluster Gutzwiller approximation, the $Δ$-Learning approach offers a computationally efficient and accurate method for studying phase transitions in large, complex bosonic systems.

DOI
10.1103/1p35-lkwv

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The cluster Gutzwiller method is widely used to study the strongly correlated bosonic systems, owing to its ability to provide a more precise description of quantum fluctuations. However, its utility is limited by the exponential increase in computational complexity as the cluster size grows. To overcome this limitation, we propose an artificial intelligence-based method known as ΔΔΔ-Learning. This approach constructs a predictive model by learning the discrepancies between lower-precision (small cluster sizes) and high-precision (large cluster sizes) implementations of the cluster Gutzwiller method, requiring only a small number of training samples. Using this predictive model, we can effectively forecast the outcomes of high-precision methods with high accuracy. Applied to various Bose-Hubbard models, the ΔΔΔ-Learning method effectively predicts phase diagrams while significantly reducing the computational resources and time. Furthermore, we have compared the predictive accuracy of ΔΔΔ-Learning with other direct learning methods and found that ΔΔΔ-Learning exhibits superior performance in scenarios with limited training data. Therefore, when combined with the cluster Gutzwiller approximation, the ΔΔΔ-Learning approach offers a computationally efficient and accurate method for studying phase transitions in large, complex bosonic systems.
A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.
Learning from data has led to a paradigm shift in computational materials science. In particular, it has been shown that neural networks can learn the potential energy surface and interatomic forces through examples, thus bypassing the computationally expensive density functional theory calculations. Combining many-body techniques with a deep-learning approach, we demonstrate that a fully connected neural network is able to learn the complex collective behavior of electrons in strongly correlated systems. Specifically, we consider the Anderson-Hubbard (AH) model, which is a canonical system for studying the interplay between electron correlation and strong localization. The ground states of the AH model on a square lattice are obtained using the real-space Gutzwiller method. The obtained solutions are used to train a multitask multilayer neural network, which subsequently can accurately predict quantities such as the local probability of double occupation and the quasiparticle weight, given the disorder potential in the neighborhood as the input.
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The identification of phases of matter is a challenging task, especially in quantum mechanics, where the complexity of the ground state appears to grow exponentially with the size of the system. Traditionally, physicists have to identify the relevant order parameters for the classification of the different phases. We here follow a radically different approach: we address this problem with a state-of-the-art deep learning technique, adversarial domain adaptation. We derive the phase diagram of the whole parameter space starting from a fixed and known subspace using unsupervised learning. This method has the advantage that the input of the algorithm can be directly the ground state without any ad hoc feature engineering. Furthermore, the dimension of the parameter space is unrestricted. More specifically, the input data set contains both labeled and unlabeled data instances. The first kind is a system that admits an accurate analytical or numerical solution, and one can recover its phase diagram. The second type is the physical system with an unknown phase diagram. Adversarial domain adaptation uses both types of data to create invariant feature extracting layers in a deep learning architecture. Once these layers are trained, we can attach an unsupervised learner to the network to find phase transitions. We show the success of this technique by applying it on several paradigmatic models: the Ising model with different temperatures, the Bose-Hubbard model, and the Su-Schrieffer-Heeger model with disorder. The method finds unknown transitions successfully and predicts transition points in close agreement with standard methods. This study opens the door to the classification of physical systems where the phase boundaries are complex such as the many-body localization problem or the Bose glass phase.
This paper examines the applicability of quantum machine learning methods to optical character recognition (OCR). The paper's abstract is given below: This paper builds on previous research demonstrating the effectiveness of quantum machine learning methods on limited or restricted data sets. A simulated quantum circuit is used to determine the validity of quantum machine learning methods for optical character recognition (OCR). OCR models are typically implemented using Convolutional Neural Networks (CNN), which can require large data sets to be trained effectively. A demonstrated ability to produce accurate OCR results with quantum machine learning would enable OCR-like models to be used when only limited data is available. This paper compares the effectiveness of quantum machine learning and classical CNN models on restricted samples of the MNIST data set, implemented in Python using IBM's Qiskit library and TensorFlow. Due to better generalization and feature space expressivity, the quantum model outperformed the CNN by an average of 6.33% on training data sizes of 10 samples to 300 samples. Both predictions had equal accuracy at N=300N=300N=300 samples, suggesting the CNN will either overtake or remain equal to the accuracy rate of the QSVC for larger data sample sizes. The space used by each model was explicitly measured, with the QSVC utilizing approximately 86% less memory space than the CNN. Upon evaluation of kernel heatmaps and eigenvalue spectra, performance improvements were minimal past N>200N>200N>200 samples for training data. The time complexity of this QSVC model has a minimum of Ω(N2)\Omega(N^2)Ω(N2), so the time cost outweighs any performance benefit. The ability to run OCR models with reliable accuracy for small data sets creates possibilities for on-device OCR models, launching new opportunities for highly specialized edge models. This will evaluated in future research.
The simulation of quantum many-body systems poses a significant challenge in physics due to the exponential scaling of Hilbert space with the number of particles. Traditional methods often struggle with large system sizes and frustrated lattices. In this research article, we present a novel algorithm that leverages the power of deep neural networks combined with Markov Chain Monte Carlo simulation to address these limitations. Our method introduces a neural network architecture specifically designed to represent bosonic quantum states on a 1D lattice chain. We successfully achieve the ground state of the Bose-Hubbard model, demonstrating the superiority of the adaptive momentum optimizer for convergence speed and stability. Notably, our approach offers flexibility in simulating various lattice geometries and potentially larger system sizes, making it a valuable tool for exploring complex quantum phenomena. This work represents a substantial advancement in the field of quantum simulation, opening new possibilities for investigating previously challenging systems.