Neural quantum states (NQS) have emerged as a powerful tool for approximating quantum wavefunctions using deep learning. While these models achieve remarkable accuracy, understanding how they encode physical information remains an open challenge. In this work, we introduce adiabatic fine-tuning, a scheme that trains NQS across a phase diagram, leading to strongly correlated weight representations across different models. This correlation in weight space enables the detection of phase transitions in quantum systems by analyzing the trained network weights alone. We validate our approach on the transverse field Ising model and the J1-J2 Heisenberg model, demonstrating that phase transitions manifest as distinct structures in weight space. Our results establish a connection between physical phase transitions and the geometry of neural network parameters, opening new directions for the interpretability of machine learning models in physics.
Neural quantum states (NQS) have emerged as a powerful tool for approximating quantum wavefunctions using deep learning. While these models achieve remarkable accuracy, understanding how they encode physical information remains an open challenge. In this work, we introduce adiabatic fine-tuning, a scheme that trains NQS across a phase diagram, leading to strongly correlated weight representations across different models. This correlation in weight space enables the detection of phase transitions in quantum systems by analyzing the trained network weights alone. We validate our approach on the transverse field Ising model and the J1-J2 Heisenberg model, demonstrating that phase transitions manifest as distinct structures in weight space. Our results establish a connection between physical phase transitions and the geometry of neural network parameters, opening new directions for the interpretability of machine learning models in physics.
Recent progress in the design and optimization of neural-network quantum states (NQSs) has made them an effective method to investigate ground-state properties of quantum many-body systems. In contrast to the standard approach of training a separate NQS from scratch at every point of the phase diagram, we demonstrate that the optimization of a NQS at a highly expressive point of the phase diagram (i.e., close to a phase transition) yields features that can be reused to accurately describe a wide region across the transition. We demonstrate the feasibility of our approach on different systems in one and two dimensions by initially pretraining a NQS at a given point of the phase diagram, followed by fine-tuning only the output layer for all other points. Notably, the computational cost of the fine-tuning step is very low compared to the pretraining stage. We argue that the reduced cost of this paradigm has significant potential to advance the exploration of strongly correlated systems using NQS, mirroring the success of fine-tuning in machine learning and natural language processing. Published by the American Physical Society 2024
Neural Quantum States (NQS) are a remarkably expressive class of variational ansätze for quantum many-body wavefunctions, yet little is understood about their internal mechanisms: trained on variational objectives alone, how do NQS accurately capture physical observables that they have never been explicitly optimized for? In this work, we present a systematic approach to analyze the internal activations of NQS using sparse autoencoders. We extract features from the residual stream and demonstrate that these features strongly correlate with physical observables such as order parameters, staggered magnetization, and half-chain correlators, across both ground state representation and real-time dynamics. Remarkably, the discovery of these features is entirely unsupervised, with no physical labels provided. We further establish that such features causally affect the corresponding observables predicted by NQS, by showing that targeted, post-training intervention on a \textit{single} feature smoothly and monotonically steers the corresponding observable, while leaving the variational energy nearly unchanged. These results demonstrate that NQS are not merely functional approximators, but encode rich, interpretable internal representations of physical information. Our approach provides both a diagnostic and an intervention tool for NQS, and serves as a foundation for using mechanistic interpretability towards more reliable, transparent NQS.
Machine learning has been applied to a wide variety of models, from classical statistical mechanics to quantum strongly correlated systems, for classifying phase transitions. The recently proposed quantum convolutional neural network (QCNN) provides a new framework for using quantum circuits instead of classical neural networks as the backbone of classification methods. We present the results from training the QCNN by the wavefunctions of the variational quantum eigensolver for the one-dimensional transverse field Ising model (TFIM). We demonstrate that the QCNN identifies wavefunctions corresponding to the paramagnetic and ferromagnetic phases of the TFIM with reasonable accuracy. The QCNN can be trained to predict the corresponding ‘phase’ of wavefunctions around the putative quantum critical point even though it is trained by wavefunctions far away. The paper provides a basis for exploiting the QCNN to identify the quantum critical point.
Neural quantum states (NQS) have emerged as a powerful variational ansatz for representing quantum many-body wave functions. Their internal mechanisms, however, remain poorly understood. We investigate the role of correlations in NQS-like quantum state representation by employing a correlation-based interpretable neural network architecture and then proving our observations using Boolean function theory. The correlator neural network demonstrates that, even for simple product states, correlations up to the systemsize in the chosen computational basis are required to represent a quantum state faithfully. We explain these observations using Fourier expansion, which reveals the correlator basis as the effective basis of the internal NQS structure, the resulting necessity for high-order correlations as supported by an entanglement bound that scales with the correlation order, the consequences of linear dependencies in constrained Hilbert spaces for correlation requirements, and the connections between spin basis rotations and the correlator basis. Furthermore, we analyze how neural networks achieve high correlation orders by increasing the magnitude of the network weights, which can be compensated by increasing the network depth. Lastly, we discuss how activation functions, network architectures, and the choice of reference basis influence correlation requirements. Our results provide insights and a better understanding of the internal structure and requirements of NQS, enabling a more systematic use of NQS in future research.
Despite the huge theoretical potential of neural quantum states, their use in describing generic, highly-correlated quantum many-body systems still often poses practical difficulties. Customized network architectures are under active investigation to address these issues. For a guided search of suited network architectures a deepened understanding of the link between neural network properties and attributes of the physical system one is trying to describe, is imperative. Drawing inspiration from the field of machine learning, in this work we show how information propagation in deep neural networks impacts the physical entanglement properties of deep neural quantum states. In fact, we link a previously identified information propagation phase transition of a neural network to a similar transition of entanglement in neural quantum states. With this bridge we can identify optimal neural quantum state hyperparameter regimes for representing area as well as volume law entangled states. The former are easily accessed by alternative methods, such as tensor network representations, at least in low physical dimensions, while the latter are challenging to describe generally due to their extensive quantum entanglement. This advance of our understanding of network configurations for accurate quantum state representation helps to develop effective representations to deal with volume-law quantum states, and we apply these findings to describe the ground state (area law state) vs. the excited state (volume law state) properties of the prototypical next-nearest neighbor spin-1/2 Heisenberg model.
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.