We employ variational autoencoders to extract physical insight from a dataset of one-particle Anderson impurity model spectral functions. Autoencoders are trained to find a low-dimensional, latent space representation that faithfully characterizes each element of the training set, as measured by a reconstruction error. Variational autoencoders, a probabilistic generalization of standard autoencoders, further condition the learned latent space to promote highly interpretable features. In our study, we find that the learned latent variables strongly correlate with well known, but nontrivial, parameters that characterize emergent behaviors in the Anderson impurity model. In particular, one latent variable correlates with particle-hole asymmetry, while another is in near one-to-one correspondence with the Kondo temperature, a dynamically generated low-energy scale in the impurity model. Using symbolic regression, we model this variable as a function of the known bare physical input parameters and ``rediscover'' the nonperturbative formula for the Kondo temperature. The machine learning pipeline we develop suggests a general purpose approach, which opens opportunities to discover new domain knowledge in other physical systems.
We employ variational autoencoders to extract physical insight from a dataset of one-particle Anderson impurity model spectral functions. Autoencoders are trained to find a low-dimensional, latent space representation that faithfully characterizes each element of the training set, as measured by a reconstruction error. Variational autoencoders, a probabilistic generalization of standard autoencoders, further condition the learned latent space to promote highly interpretable features. In our study, we find that the learned latent variables strongly correlate with well known, but nontrivial, parameters that characterize emergent behaviors in the Anderson impurity model. In particular, one latent variable correlates with particle-hole asymmetry, while another is in near one-to-one correspondence with the Kondo temperature, a dynamically generated low-energy scale in the impurity model. Using symbolic regression, we model this variable as a function of the known bare physical input parameters and ``rediscover'' the nonperturbative formula for the Kondo temperature. The machine learning pipeline we develop suggests a general purpose approach, which opens opportunities to discover new domain knowledge in other physical systems.
We employ variational autoencoders to extract physical insight from a dataset of one-particle Anderson impurity model spectral functions. Autoencoders are trained to find a low-dimensional, latent space representation that faithfully characterizes each element of the training set, as measured by a reconstruction error. Variational autoencoders, a probabilistic generalization of standard autoencoders, further condition the learned latent space to promote highly interpretable features. In our study, we find that the learned latent space components strongly correlate with well known, but nontrivial, parameters that characterize emergent behaviors in the Anderson impurity model. In particular, one latent space component correlates with particle-hole asymmetry, while another is in near one-to-one correspondence with the Kondo temperature, a dynamically generated low-energy scale in the impurity model. With symbolic regression, we model this component as a function of bare physical input parameters and rediscover the non-perturbative formula for the Kondo temperature. The machine learning pipeline we develop opens opportunities to discover new domain knowledge in other physical systems.
Analog quantum simulators provide access to many-body dynamics beyond the reach of classical computation. However, extracting physical insights from experimental data is often hindered by measurement noise, limited observables, and incomplete knowledge of the underlying microscopic model. Here, we develop a machine learning approach based on a variational autoencoder (VAE) to analyze interference measurements of tunnel-coupled one-dimensional Bose gases, which realize the sine-Gordon quantum field theory. Trained in an unsupervised manner, the VAE learns a minimal latent representation that strongly correlates with the equilibrium control parameter of the system. Applied to nonequilibrium protocols, the latent space uncovers signatures of frozen-in solitons following rapid cooling, and reveals anomalous postquench dynamics not captured by conventional correlation-based methods. These results demonstrate that generative models can extract physically interpretable variables directly from noisy and sparse experimental data, providing complementary probes of equilibrium and nonequilibrium physics in quantum simulators. More broadly, our work highlights how machine learning can supplement established field-theoretical techniques, paving the way for scalable, data-driven discovery in quantum many-body systems.
Variational autoencoders are powerful representation learning models that map complex data into low-dimensional latent spaces, enabling the discovery of interpretable and disentangled factors. Such representations can facilitate the interpretation and controllable generation of data describing complex scientific systems. Understanding how these factors are organized and encoded in latent space is therefore important for developing reliable representation learning models. Recently, quantum variational autoencoders (QVAEs) have been proposed as quantum representation models, demonstrating informative latent representations and improved latent-space occupancy through quantum regularization. However, it remains unclear whether and how QVAEs can learn disentangled and interpretable latent factors. A key challenge in investigating quantum latent factors is that a small number of qubits spans an exponentially large Hilbert space, making the notion of an individual quantum latent dimension nontrivial. Here, we investigate what constitutes an individual quantum latent dimension and whether it can encode a distinct factor. We develop theoretical insights into quantum latent dimensions and support them with empirical studies on representative synthetic problems, including MNIST variants. Across three datasets, we demonstrate that QVAEs can discover factorized and semantically interpretable latent representations, with individual qubits functioning as meaningful latent factors. These results establish a foundation for understanding quantum latent spaces and their potential for structured and interpretable representation learning.
Interpretable machine learning is rapidly becoming a crucial tool for scientific discovery. Among existing approaches, variational autoencoders (VAEs) have shown promise in extracting the hidden physical features of some input data, with no supervision nor prior knowledge of the system at study. Yet the ability of VAEs to create meaningful, interpretable representations relies on their accurate approximation of the underlying probability distribution of their input. When dealing with quantum data, VAEs must hence account for its intrinsic randomness and complex correlations. While VAEs have been previously applied to quantum data, they have often neglected its probabilistic nature, hindering the extraction of meaningful physical descriptors. Here we demonstrate that two key modifications enable VAEs to learn physically meaningful latent representations: a decoder capable of faithfully reproducing the probability distributions of quantum states and a probabilistic loss tailored to this task. Using benchmark quantum spin models, we identify regimes where standard methods fail while the representations learned by our approach remain meaningful and interpretable. Applied to experimental data from Rydberg atom arrays, the model autonomously uncovers the phase structure without access to prior labels, Hamiltonian details, or knowledge of relevant order parameters, highlighting its potential as an unsupervised and interpretable tool for the study of quantum systems.
Interpretable machine learning techniques are becoming essential tools for extracting physical insights from complex quantum data. We build on recent advances in variational autoencoders to demonstrate that such models can learn physically meaningful and interpretable representations from a broad class of unlabeled quantum datasets. From raw measurement data alone, the learned representation reveals rich information about the underlying structure of quantum phase diagrams. We further augment the learning pipeline with symbolic methods, enabling the discovery of compact analytical descriptors that serve as order parameters for the distinct regimes emerging in the learned representations. We demonstrate the framework on experimental Rydberg-atom snapshots, classical shadows of the cluster Ising model, and hybrid discrete-continuous fermionic data, revealing previously unreported phenomena such as a corner-ordering pattern in the Rydberg arrays. These results establish a general framework, which constitutes an important step toward the autonomous discovery of physical descriptors from diverse quantum datasets. All methods are available through qdisc, an open-source Python library designed to make these tools accessible to the broader community.
Generalized quantum impurity models—which feature a few localized and strongly correlated degrees of freedom coupled to itinerant conduction electrons—describe diverse physical systems, from magnetic moments in metals to nanoelectronics quantum devices such as quantum dots or single-molecule transistors. Correlated materials can also be understood as self-consistent impurity models through dynamical mean-field theory. Accurate simulation of such models is challenging, especially at low temperatures, due to many-body effects from electronic interactions, resulting in strong renormalization. In particular, the interplay between local impurity complexity and Kondo physics is highly nontrivial. A common approach, which we further develop in this work, is to consider instead a simpler effective impurity model that still captures the low-energy physics of interest. The mapping from a bare to an effective model is typically done perturbatively, but even this can be difficult for complex systems, and the resulting effective model parameters can nevertheless be quite inaccurate. Here we develop a nonperturbative, unsupervised machine learning approach to systematically obtain low-energy effective impurity-type models, based on the renormalization-group framework. The method is shown to be general and flexible, as well as accurate and systematically improvable. We benchmark the method against exact results for the Anderson impurity model, and we provide an outlook for more complex models beyond the reach of existing methods. Published by the American Physical Society 2024