Abstract Theoretical approaches to quantum many-body physics require developing compact representations of the complexity of generic quantum states. This paper explores an interpretable data-driven approach utilizing principal component analysis (PCA) and autoencoder neural networks to compress the two-particle vertex, a key element in Feynman diagram approaches. We show that the linear PCA offers more physical insight and better out-of-distribution generalization than the nominally more expressive autoencoders. Even with ∼10–20 principal components, we find excellent reconstruction across the phase diagram suggesting the existence of heretofore unrealized structures in the diagrammatic theory. We show that the principal components needed to describe the ferromagnetic state are not contained in the low rank description of the Fermi liquid (FL) state, unlike those for antiferromagnetic and superconducting states, suggesting that the latter two states emerge from pre-existing fluctuations in the FL while ferromagnetism is driven by a different process.
Abstract Theoretical approaches to quantum many-body physics require developing compact representations of the complexity of generic quantum states. This paper explores an interpretable data-driven approach utilizing principal component analysis (PCA) and autoencoder neural networks to compress the two-particle vertex, a key element in Feynman diagram approaches. We show that the linear PCA offers more physical insight and better out-of-distribution generalization than the nominally more expressive autoencoders. Even with ∼10–20 principal components, we find excellent reconstruction across the phase diagram suggesting the existence of heretofore unrealized structures in the diagrammatic theory. We show that the principal components needed to describe the ferromagnetic state are not contained in the low rank description of the Fermi liquid (FL) state, unlike those for antiferromagnetic and superconducting states, suggesting that the latter two states emerge from pre-existing fluctuations in the FL while ferromagnetism is driven by a different process.
Thermodynamics is a theory of principles that permits a basic description of the macroscopic properties of a rich variety of complex systems from traditional ones, such as crystalline solids, gases, liquids, and thermal machines, to more intricate systems such as living organisms and black holes to name a few. Physical quantities of interest, or equilibrium state variables, are linked together in equations of state to give information on the studied system, including phase transitions, as energy in the forms of work and heat, and/or matter are exchanged with its environment, thus generating entropy. A more accurate description requires different frameworks, namely, statistical mechanics and quantum physics to explore in depth the microscopic properties of physical systems and relate them to their macroscopic properties. These frameworks also allow to go beyond equilibrium situations. Given the notably increasing complexity of mathematical models to study realistic systems, and their coupling to their environment that constrains their dynamics, both analytical approaches and numerical methods that build on these models show limitations in scope or applicability. On the other hand, machine learning, i.e., data-driven, methods prove to be increasingly efficient for the study of complex quantum systems. Deep neural networks, in particular, have been successfully applied to many-body quantum dynamics simulations and to quantum matter phase characterization. In the present work, we show how to use a variational autoencoder (VAE)—a state-of-the-art tool in the field of deep learning for the simulation of probability distributions of complex systems. More precisely, we transform a quantum mechanical problem of many-body state reconstruction into a statistical problem, suitable for VAE, by using informationally complete positive operator-valued measure. We show, with the paradigmatic quantum Ising model in a transverse magnetic field, that the ground-state physics, such as, e.g., magnetization and other mean values of observables, of a whole class of quantum many-body systems can be reconstructed by using VAE learning of tomographic data for different parameters of the Hamiltonian, and even if the system undergoes a quantum phase transition. We also discuss challenges related to our approach as entropy calculations pose particular difficulties.
This paper evaluates the performance of two quantum machine learning algorithms: Quantum Support Vector Machine (QSVM) and Quantum Principal Component Analysis (Q-PCA). The scalability of QSVM is analyzed across increasing dataset sizes and feature dimensions, revealing faster-than-linear growth in kernel evaluation and SVM training times. However, this computational overhead is accompanied by a notable decline in classification accuracy, highlighting challenges in applying QSVM to larger or more complex datasets. In contrast, Q-PCA demonstrates near-linear scalability with dataset size and qubit count, leveraging the independence of quantum circuit execution and structured operations to maintain efficiency. A QNN-based implementation of Q-PCA further validates its ability to align data with principal components, maximize variance, and enable effective dimensionality reduction. These findings emphasize the potential of Q-PCA for efficient quantum-enhanced data analysis while underscoring the need for optimization in QSVM and further exploration of Q-PCA’s capabilities for real-world quantum computing applications.
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.
Abstract Machine learning techniques have been widely used in the study of strongly correlated systems in recent years. Here, we review some applications to classical and quantum many-body systems and present results from an unsupervised machine learning technique, the principal component analysis, employed to identify the finite-temperature phase transition of the three-dimensional Fermi-Hubbard model to the antiferromagnetically ordered state. We find that this linear method can capture the phase transition as well as other more complicated and nonlinear counterparts.
We introduce an unsupervised machine-learning framework that discovers optimally compressed representations of quantum many-body ground states. Using an autoencoder neural network architecture on data from L-site Fermi-Hubbard models, we identify minimal latent spaces with a sharp reconstruction quality threshold at L−1 latent dimensions, matching the system's intrinsic degrees of freedom. We demonstrate the use of the trained decoder as a differentiable variational ansatz to minimize energy directly within the latent space. Crucially, this approach circumvents the N-representability problem, as the learned manifold implicitly restricts the optimization to physically valid quantum states.
Quantum neural networks (QNNs) offer a promising avenue for machine learning by leveraging quantum computation, but practical QNN models are constrained by the low dimensionality of the input data they can handle. This paper proposes a hybrid quantum-classical neural network architecture that incorporates principal component analysis (PCA) for dimensionality reduction and variational quantum circuits (VQC) for quantum feature encoding. Focusing on the Wisconsin Diagnostic Breast Cancer dataset (569 instances, 30 features) as a representative biomedical classification task, in this work, we design hybrid quantum-classical neural network (QNN) models using 5-qubit and 10-qubit quantum circuits. For comparison, we use classical baseline identical to the downstream head but without the VQC+measurement. The preprocessed inputs feed the classifier head directly; only the input layer dimension is adjusted to the number of features. The 5-qubit hybrid model is evaluated under two conditions: with and without principal component analysis (PCA) applied as a preprocessing step to investigate the effect of dimensionality reduction on model performance. In contrast, the 10 -qubit hybrid model is evaluated without PCA, directly processing the input features to explore its capacity to handle higher-dimensional data within the quantum circuit. Performance is measured in terms of classification accuracy, precision, recall, and F1-score. We evaluate the model’s robustness by injecting noise into the training data, finding that the PCA-enhanced hybrid QNN maintains competitive accuracy and resilience under perturbation. This study highlights the potential of PCA-driven variational quantum encodings to enhance QNNs for reliable biomedical data classification.