Machine learning quantum phases of matter beyond the fermion sign problem

Machine learning assisted plasmonic metascreen for enhanced broadband absorption in ultra-thin silicon films

We propose and demonstrate a data-driven plasmonic metascreen that efficiently absorbs incident light over a wide spectral range in an ultra-thin silicon film. By embedding a double-nanoring silver array within a 20 nm ultrathin amorphous silicon (a-Si) layer, we achieve a significant enhancement of light absorption. This enhancement arises from the interaction between the resonant cavity modes and localized plasmonic modes, requiring precise tuning of plasmon resonances to match the absorption region of the silicon active layer. To facilitate the device design and improve light absorption without increasing the thickness of the active layer, we develop a deep learning framework, which learns to map from the absorption spectra to the design space. This inverse design strategy helps to tune the absorption for selective spectral functionalities. Our optimized design surpasses the bare silicon planar device, exhibiting a remarkable enhancement of over 100%. Experimental validation confirms the broadband enhancement of light absorption in the proposed configuration. The proposed metascreen absorber holds great potential for light harvesting applications and may be leveraged to improve the light conversion efficiency of ultra-thin silicon solar cells, photodetectors, and optical filters.

Gaussian processes for finite size extrapolation of many-body simulations

Key to being able to accurately model the properties of realistic materials is being able to predict their properties in the thermodynamic limit. Nevertheless, because most many-body electronic structure methods scale as a high-order polynomial, or even exponentially, with system size, directly simulating large systems in their thermodynamic limit rapidly becomes computationally intractable. As a result, researchers typically estimate the properties of large systems that approach the thermodynamic limit by extrapolating the properties of smaller, computationally-accessible systems based on relatively simple scaling expressions. In this work, we employ Gaussian processes to more accurately and efficiently extrapolate many-body simulations to their thermodynamic limit. We train our Gaussian processes on Smooth Overlap of Atomic Positions (SOAP) descriptors to extrapolate the energies of one-dimensional hydrogen chains obtained using two high-accuracy many-body methods: coupled cluster theory and Auxiliary Field Quantum Monte Carlo (AFQMC). In so doing, we show that Gaussian processes trained on relatively short 10-30-atom chains can predict the energies of both homogeneous and inhomogeneous hydrogen chains in their thermodynamic limit with sub-milliHartree accuracy. Unlike standard scaling expressions, our GPR-based approach is highly generalizable given representative training data and is not dependent on systems' geometries or dimensionality. This work highlights the potential for machine learning to correct for the finite size effects that routinely complicate the interpretation of finite size many-body simulations.

Unsupervised representation learning of Kohn-Sham states and consequences for downstream predictions of many-body effects

Representation learning for the electronic structure problem is a major challenge of machine learning in computational condensed matter and materials physics. Within quantum mechanical first principles approaches, density functional theory (DFT) is the preeminent tool for understanding electronic structure, and the high-dimensional DFT wavefunctions serve as building blocks for downstream calculations of correlated many-body excitations and related physical observables. Here, we use variational autoencoders (VAE) for the unsupervised learning of DFT wavefunctions and show that these wavefunctions lie in a low-dimensional manifold within latent space. Our model autonomously determines the optimal representation of the electronic structure, avoiding limitations due to manual feature engineering. To demonstrate the utility of the latent space representation of the DFT wavefunction, we use it for the supervised training of neural networks (NN) for downstream prediction of quasiparticle bandstructures within the GW formalism. The GW prediction achieves a low error of 0.11 eV for a combined test set of two-dimensional metals and semiconductors, suggesting that the latent space representation captures key physical information from the original data. Finally, we explore the generative ability and interpretability of the VAE representation.

Study of the Berezinskii-Kosterlitz-Thouless transition: an unsupervised machine learning approach

The Berezinskii-Kosterlitz-Thouless (BKT) transition in magnetic systems is an intriguing phenomenon, and estimating the BKT transition temperature is a long-standing problem. In this work, we explore anisotropic classical Heisenberg XY and XXZ models with ferromagnetic exchange on a square lattice and antiferromagnetic exchange on a triangular lattice using an unsupervised machine learning approach called principal component analysis (PCA). The earlier PCA studies of the BKT transition temperature (TBKT) using the vorticities as input fail to give any conclusive results, whereas, in this work, we show that the proper analysis of the first principal component-temperature curve can estimateTBKTwhich is consistent with the existing literature. This analysis works well for the anisotropic classical Heisenberg with a ferromagnetic exchange on a square lattice and for frustrated antiferromagnetic exchange on a triangular lattice. The classical anisotropic Heisenberg antiferromagnetic model on the triangular lattice has two close transitions: theTBKTand Ising-like phase transition for chirality atTc, and it is difficult to separate these transition points. It is also noted that using the PCA method and manipulation of their first principal component not only makes the separation of transition points possible but also determines transition temperature.

Machine learning analysis of dimensional reduction conjecture for nonequilibrium Berezinskii-Kosterlitz-Thouless transition in three dimensions

We investigate the recently proposed dimensional reduction conjecture in driven disordered systems using a machine learning technique. The conjecture states that a static snapshot of a disordered system driven at a constant velocity is equal to a space-time trajectory of its lower-dimensional pure counterpart. This suggests that the three-dimensional (3D) random field XY model exhibits the Berezinskii-Kosterlitz-Thouless transition when driven out of equilibrium. To verify the conjecture directly by observing configurations of the system, we utilize the capacity of neural networks to detect subtle features of images. Specifically, we train neural networks to differentiate snapshots of the 3D driven random field XY model from space-time trajectories of the two-dimensional pure XY model. Our results demonstrate that the network cannot distinguish between the two, confirming the dimensional reduction conjecture.

Understanding quantum machine learning also requires rethinking generalization

Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.

Transfer learning from Hermitian to non-Hermitian quantum many-body physics

Identifying phase boundaries of interacting systems is one of the key steps to understanding quantum many-body models. The development of various numerical and analytical methods has allowed exploring the phase diagrams of many Hermitian interacting systems. However, numerical challenges and scarcity of analytical solutions hinder obtaining phase boundaries in non-Hermitian many-body models. Recent machine learning methods have emerged as a potential strategy to learn phase boundaries from various observables without having access to the full many-body wavefunction. Here, we show that a machine learning methodology trained solely on Hermitian correlation functions allows identifying phase boundaries of non-Hermitian interacting models. These results demonstrate that Hermitian machine learning algorithms can be redeployed to non-Hermitian models without requiring further training to reveal non-Hermitian phase diagrams. Our findings establish transfer learning as a versatile strategy to leverage Hermitian physics to machine learning non-Hermitian phenomena.

Model-independent quantum phases classifier

Machine learning has transformed science and technology. In this article, we present a model-independent classifier that uses the k-Nearest Neighbors algorithm to classify phases of a model for which it has never been trained. This is done by studying three different spin-1 chains with some common phases: the XXZ chains with uniaxial single-ion-type anisotropy, the bond alternating XXZ chains, and the bilinear biquadratic chain. We show that the algorithm trained with two of these models can, with high probability, determine phases common to the third one. This is the first step towards a universal classifier, where an algorithm can recognize an arbitrary phase without knowing the Hamiltonian, since it knows only partial information about the quantum state.

Berezinskii-Kosterlitz-Thouless transition from neural network flows

We adopt the neural network (NN) flow method to study the Berezinskii-Kosterlitz-Thouless (BKT) phase transitions of the two-dimensional q-state clock model with q≥4. The NN flow consists of a sequence of the same units that proceed with the flow. This unit is a variational autoencoder trained by the data of Monte Carlo configurations in unsupervised learning. To gauge the difference among the ensembles of Monte Carlo configurations at different temperatures and the uniqueness of the ensemble of NN-flow states, we adopt the Jensen-Shannon divergence (JSD) as the information-distance measure "thermometer." This JSD thermometer compares the probability distribution functions of the mean spin value of two ensembles of states. Our results show that the NN flow will flow an arbitrary spin state to some state in a fixed-point ensemble of states. The corresponding JSD of the fixed-point ensemble takes a unique profile with peculiar features, which can help to identify the critical temperatures of BKT phase transitions of the underlying Monte Carlo configurations.

Learning by confusion approach to identification of discontinuous phase transitions

Recently, the learning by confusion (LbC) approach has been proposed as a machine learning tool to determine the critical temperature T_{c} of phase transitions without any prior knowledge of its even approximate value. The method has been proven effective, but it has been used only for continuous phase transitions, where the confusion results only from deliberate incorrect labeling of the data. However, in the case of a discontinuous phase transition, additional confusion can result from the coexistence of different phases. To verify whether the confusion scheme can also be used for discontinuous phase transitions, we apply the LbC method to three microscopic models, the Blume-Capel, the q-state Potts, and the Falicov-Kimball models, which undergo continuous or discontinuous phase transitions depending on model parameters. With the help of a simple model, we predict that the phase coexistence present in discontinuous phase transitions can indeed make the neural network more confused and thus decrease its performance. However, numerical calculations performed for the models mentioned above indicate that other aspects of this kind of phase transition are more important and can render the LbC method even less effective. Nevertheless, we demonstrate that in some cases the same aspects allow us to use the LbC method to identify the order of a phase transition.

Boundary between noise and information applied to filtering neural network weight matrices

Deep neural networks have been successfully applied to a broad range of problems where overparametrization yields weight matrices which are partially random. A comparison of weight matrix singular vectors to the Porter-Thomas distribution suggests that there is a boundary between randomness and learned information in the singular value spectrum. Inspired by this finding, we introduce an algorithm for noise filtering, which both removes small singular values and reduces the magnitude of large singular values to counteract the effect of level repulsion between the noise and the information part of the spectrum. For networks trained in the presence of label noise, we indeed find that the generalization performance improves significantly due to noise filtering.

Super-resolution of magnetic systems using deep learning

We construct a deep neural network to enhance the resolution of spin structure images formed by spontaneous symmetry breaking in the magnetic systems. Through the deep neural network, an image is expanded to a super-resolution image and reduced to the original image size to be fitted with the input feed image. The network does not require ground truth images in the training process. Therefore, it can be applied when low-resolution images are provided as training datasets, while high-resolution images are not obtainable due to the intrinsic limitation of microscope techniques. To show the usefulness of the network, we train the network with two types of simulated magnetic structure images; one is from self-organized maze patterns made of chiral magnetic structures, and the other is from magnetic domains separated by walls that are topological defects of the system. The network successfully generates high-resolution images highly correlated with the exact solutions in both cases. To investigate the effectiveness and the differences between datasets, we study the network's noise tolerance and compare the networks' reliabilities. The network is applied with experimental data obtained by magneto-optical Kerr effect microscopy and spin-polarized low-energy electron microscopy.

Exploring nonlinear dynamics and network structures in Kuramoto systems using machine learning approaches

Recent advances in machine learning (ML) have facilitated its application to a wide range of systems, from complex to quantum. Reservoir computing algorithms have proven particularly effective for studying nonlinear dynamical systems that exhibit collective behaviors, such as synchronizations and chaotic phenomena, some of which still remain unclear. Here, we apply ML approaches to the Kuramoto model to address several intriguing problems, including identifying the transition point and criticality of a hybrid synchronization transition, predicting future chaotic behaviors, and understanding network structures from chaotic patterns. Our proposed method also has further implications, such as inferring the structure of neural networks from electroencephalogram signals. This study, finally, highlights the potential of ML approaches for advancing our understanding of complex systems.

Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks

Quantum convolutional neural networks (QCNNs) have been introduced as classifiers for gapped quantum phases of matter. Here, we propose a model-independent protocol for training QCNNs to discover order parameters that are unchanged under phase-preserving perturbations. We initiate the training sequence with the fixed-point wave functions of the quantum phase and add translation-invariant noise that respects the symmetries of the system to mask the fixed-point structure on short length scales. We illustrate this approach by training the QCNN on phases protected by time-reversal symmetry in one dimension, and test it on several time-reversal symmetric models exhibiting trivial, symmetry-breaking, and symmetry-protected topological order. The QCNN discovers a set of order parameters that identifies all three phases and accurately predicts the location of the phase boundary. The proposed protocol paves the way toward hardware-efficient training of quantum phase classifiers on a programmable quantum processor.

Group-equivariant autoencoder for identifying spontaneously broken symmetries

We introduce the group-equivariant autoencoder (GE autoencoder), a deep neural network (DNN) method that locates phase boundaries by determining which symmetries of the Hamiltonian have spontaneously broken at each temperature. We use group theory to deduce which symmetries of the system remain intact in all phases, and then use this information to constrain the parameters of the GE autoencoder such that the encoder learns an order parameter invariant to these "never-broken" symmetries. This procedure produces a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. We include symmetry regularization terms in the loss function of the GE autoencoder so that the learned order parameter is also equivariant to the remaining symmetries of the system. By examining the group representation by which the learned order parameter transforms, we are then able to extract information about the associated spontaneous symmetry breaking. We test the GE autoencoder on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE autoencoder (1) accurately determines which symmetries have spontaneously broken at each temperature; (2) estimates the critical temperature in the thermodynamic limit with greater accuracy, robustness, and time efficiency than a symmetry-agnostic baseline autoencoder; and (3) detects the presence of an external symmetry-breaking magnetic field with greater sensitivity than the baseline method. Finally, we describe various key implementation details, including a quadratic-programming-based method for extracting the critical temperature estimate from trained autoencoders and calculations of the DNN initialization and learning rate settings required for fair model comparisons.

Machine learning approach to study quantum phase transitions of a frustrated one dimensional spin-1/2 system

Frustration-driven quantum fluctuation leads to many exotic phases in the ground state (GS) and the study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. We study a frustrated HeisenbergJ1-J2model of spin-1/2 chain with nearest exchange interactionJ₁and next nearest exchange interactionJ₂using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSCs) of GS and first excited state (FES) for differentJ2/J1are used as the input in PCA to construct the covariance matrix. The 'quantified principal component'p1(J2/J1)of the largest eigenvalue of the covariance matrix is calculated and it is shown that the nature and variation ofp1(J2/J1)can accurately predict the phase transitions and degeneracies in the GS. Thep1(J2/J1)calculated from the MPSC of GS only exhibits the signature of degeneracies in the GS, whereas,p1(J2/J1)calculated from the MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that the jump inp1(J2/J1)of FES atJ2/J1≈0.241, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of the first two principal components of FES shows distinct band formation for different phases. The MPSCs are obtained by using an iterative variational method (IVM) which gives the approximate eigenvalues and eigenvectors.

Random matrix analysis of deep neural network weight matrices

Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the weight matrices of trained deep neural networks using methods from random matrix theory (RMT) and show that the statistics of most of the singular values follow universal RMT predictions. This suggests that they are random and do not contain system specific information, which we investigate further by comparing the statistics of eigenvector entries to the universal Porter-Thomas distribution. We find that for most eigenvectors the hypothesis of randomness cannot be rejected, and that only eigenvectors belonging to the largest singular values deviate from the RMT prediction, indicating that they may encode learned information. In addition, a comparison with RMT predictions also allows to distinguish networks trained in different learning regimes-from lazy to rich learning. We analyze the spectral distribution of the large singular values using the Hill estimator and find that the distribution cannot in general be characterized by a tail index, i.e., is not of power-law type.

Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer

The phenomena of quantum criticality underlie many novel collective phenomena found in condensed matter systems. They present a challenge for classical and quantum simulation, in part because of diverging correlation lengths and consequently strong finite-size effects. Tensor network techniques that work directly in the thermodynamic limit can negotiate some of these difficulties. Here, we optimise a translationally invariant, sequential quantum circuit on a superconducting quantum device to simulate the groundstate of the quantum Ising model through its quantum critical point. We further demonstrate how the dynamical quantum critical point found in quenches of this model across its quantum critical point can be simulated. Our approach avoids finite-size scaling effects by using sequential quantum circuits inspired by infinite matrix product states. We provide efficient circuits and a variety of error mitigation strategies to implement, optimise and time-evolve these states.

Feed-forward neural network based variational wave function for the fermionic Hubbard model in one dimension

We explore the suitability of a feed-forward neural network (FNN) to represent the ground state of the fermionic Hubbard model in one dimension (1D). We consider the model at half-filling, represent the ground state wave function in terms of an FNN and optimize it using the variational Monte Carlo (VMC) method. The results are compared with the exact Bethe Ansatz solution. We find that for lattice sizes which give a 'filled-shell' condition for the non-interacting Fermi sea wave function, a simple FNN performs very well at all values of Hubbard interactionU. For lattice sizes where this condition is not obtained, the simple FNN fails and we find a modified network with a 'sign' component (sFNN) to work in such cases. On the flip side, though we find the FNN to be successful in providing an unbiased variational wave function for the fermionic many-body system in 1D, the computational cost for the wave function scales up rapidly with lattice size which limits its applicability.

Hessian-based toolbox for reliable and interpretable machine learning in physics

Machine learning (ML) techniques applied to quantum many-body physics have emerged as a new research field. While the numerical power of this approach is undeniable, the most expressive ML algorithms, such as neural networks, are black boxes: The user does neither know the logic behind the model predictions nor the uncertainty of the model predictions. In this work, we present a toolbox for interpretability and reliability, agnostic of the model architecture. In particular, it provides a notion of the influence of the input data on the prediction at a given test point, an estimation of the uncertainty of the model predictions, and an extrapolation score for the model predictions. Such a toolbox only requires a single computation of the Hessian of the training loss function. Our work opens the road to the systematic use of interpretability and reliability methods in ML applied to physics and, more generally, science.

Learning a compass spin model with neural network quantum states

Neural network quantum states provide a novel representation of the many-body states of interacting quantum systems and open up a promising route to solve frustrated quantum spin models that evade other numerical approaches. Yet its capacity to describe complex magnetic orders with large unit cells has not been demonstrated, and its performance in a rugged energy landscape has been questioned. Here we apply restricted Boltzmann machines (RBMs) and stochastic gradient descent to seek the ground states of a compass spin model on the honeycomb lattice, which unifies the Kitaev model, Ising model and the quantum 120° model with a single tuning parameter. We report calculation results on the variational energy, order parameters and correlation functions. The phase diagram obtained is in good agreement with the predictions of tensor network ansatz, demonstrating the capacity of RBMs in learning the ground states of frustrated quantum spin Hamiltonians. The limitations of the calculation are discussed. A few strategies are outlined to address some of the challenges in machine learning frustrated quantum magnets.

Towards ML-Based Diagnostics of Laser-Plasma Interactions

The power of machine learning (ML) in feature identification can be harnessed for determining quantities in experiments that are difficult to measure directly. However, if an ML model is trained on simulated data, rather than experimental results, the differences between the two can pose an obstacle to reliable data extraction. Here we report on the development of ML-based diagnostics for experiments on high-intensity laser-matter interactions. With the intention to accentuate robust, physics-governed features, the presence of which is tolerant to such differences, we test the application of principal component analysis, data augmentation and training with data that has superimposed noise of gradually increasing amplitude. Using synthetic data of simulated experiments, we identify that the approach based on the noise of increasing amplitude yields the most accurate ML models and thus is likely to be useful in similar projects on ML-based diagnostics.

Learning impurity spectral functions from density of states

Using numerical renormalization group calculation, we construct a dataset with 100 K samples, and train six different neural networks for the prediction of spectral functions from density of states (DOS) of the host material. We find that a combination of gated recurrent unit (GRU) network and bidirectional GRU (BiGRU) performances the best among all the six neural networks. The mean absolute error of the GRU + BiGRU network can reach 0.052 and 0.043 when this network is evaluated on the original dataset and two other independent datasets. The average time of spectral function predictions from machine learning is on the scale of 10^-5-10^-6that of traditional impurity solvers for Anderson impurity model. This investigation pave the way for the application of recurrent neural network and convolutional neural network in the prediction of spectral functions from DOSs in machine learning solvers of magnetic impurity problems.

Ab Initio Machine Learning in Chemical Compound Space

Chemical compound space (CCS), the set of all theoretically conceivable combinations of chemical elements and (meta-)stable geometries that make up matter, is colossal. The first-principles based virtual sampling of this space, for example, in search of novel molecules or materials which exhibit desirable properties, is therefore prohibitive for all but the smallest subsets and simplest properties. We review studies aimed at tackling this challenge using modern machine learning techniques based on (i) synthetic data, typically generated using quantum mechanics based methods, and (ii) model architectures inspired by quantum mechanics. Such Quantum mechanics based Machine Learning (QML) approaches combine the numerical efficiency of statistical surrogate models with an ab initio view on matter. They rigorously reflect the underlying physics in order to reach universality and transferability across CCS. While state-of-the-art approximations to quantum problems impose severe computational bottlenecks, recent QML based developments indicate the possibility of substantial acceleration without sacrificing the predictive power of quantum mechanics.

Identification of Non-Fermi Liquid Physics in a Quantum Critical Metal via Quantum Loop Topography

Non-Fermi liquid physics is ubiquitous in strongly correlated metals, manifesting itself in anomalous transport properties, such as a T-linear resistivity in experiments. However, its theoretical understanding in terms of microscopic models is lacking, despite decades of conceptual work and attempted numerical simulations. Here we demonstrate that a combination of sign-problem-free quantum Monte Carlo sampling and quantum loop topography, a physics-inspired machine-learning approach, can map out the emergence of non-Fermi liquid physics in the vicinity of a quantum critical point (QCP) with little prior knowledge. Using only three parameter points for training the underlying neural network, we are able to robustly identify a stable non-Fermi liquid regime tracing the fans of metallic QCPs at the onset of both spin-density wave and nematic order. In particular, we establish for the first time that a spin-density wave QCP commands a wide fan of non-Fermi liquid region that funnels into the quantum critical point. Our study thereby provides an important proof-of-principle example that new physics can be detected via unbiased machine-learning approaches.

Unsupervised machine learning of topological phase transitions from experimental data

Identifying phase transitions is one of the key challenges in quantum many-body physics. Recently, machine learning methods have been shown to be an alternative way of localising phase boundaries from noisy and imperfect data without the knowledge of the order parameter. Here, we apply different unsupervised machine learning techniques, including anomaly detection and influence functions, to experimental data from ultracold atoms. In this way, we obtain the topological phase diagram of the Haldane model in a completely unbiased fashion. We show that these methods can successfully be applied to experimental data at finite temperatures and to the data of Floquet systems when post-processing the data to a single micromotion phase. Our work provides a benchmark for the unsupervised detection of new exotic phases in complex many-body systems.

Screening potential topological insulators in half-Heusler compounds via compressed-sensing

Ternary half-Heusler compounds with widely tunable electronic structures, present a new platform to discover topological insulators (TIs). Due to time-consuming computations and synthesis procedures, the identification of new TIs is however a rough task. Here, we adopt a compressed-sensing approach to rapidly screen potential TIs in half-Heusler family, which is realized via a two-dimensional descriptor that only depends on the fundamental properties of the constituent atoms. Beyond the finite training data, the proposed descriptor is employed to screen many new half-Heusler compounds, including those with integer and fractional stoichiometry, and a larger number of possible TIs are predicted.

Supervised and unsupervised learning of directed percolation

Machine learning (ML) has been well applied to studying equilibrium phase transition models by accurately predicating critical thresholds and some critical exponents. Difficulty will be raised, however, for integrating ML into nonequilibrium phase transitions. The extra dimension in a given nonequilibrium system, namely time, can greatly slow down the procedure toward the steady state. In this paper we find that by using some simple techniques of ML, non-steady-state configurations of directed percolation (DP) suffice to capture its essential critical behaviors in both (1+1) and (2+1) dimensions. With the supervised learning method, the framework of our binary classification neural networks can identify the phase transition threshold, as well as the spatial and temporal correlation exponents. The characteristic time t_{c}, specifying the transition from active phases to absorbing ones, is also a major product of the learning. Moreover, we employ the convolutional autoencoder, an unsupervised learning technique, to extract dimensionality reduction representations and cluster configurations of (1+1) bond DP. It is quite appealing that such a method can yield a reasonable estimation of the critical point.

Machine learning approach to muon spectroscopy analysis

In recent years, artificial intelligence techniques have proved to be very successful when applied to problems in physical sciences. Here we apply an unsupervised machine learning (ML) algorithm called principal component analysis (PCA) as a tool to analyse the data from muon spectroscopy experiments. Specifically, we apply the ML technique to detect phase transitions in various materials. The measured quantity in muon spectroscopy is an asymmetry function, which may hold information about the distribution of the intrinsic magnetic field in combination with the dynamics of the sample. Sharp changes of shape of asymmetry functions-measured at different temperatures-might indicate a phase transition. Existing methods of processing the muon spectroscopy data are based on regression analysis, but choosing the right fitting function requires knowledge about the underlying physics of the probed material. Conversely, PCA focuses on small differences in the asymmetry curves and works without any prior assumptions about the studied samples. We discovered that the PCA method works well in detecting phase transitions in muon spectroscopy experiments and can serve as an alternative to current analysis, especially if the physics of the studied material are not entirely known. Additionally, we found out that our ML technique seems to work best with large numbers of measurements, regardless of whether the algorithm takes data only for a single material or whether the analysis is performed simultaneously for many materials with different physical properties.

A cautionary tale for machine learning generated configurations in presence of a conserved quantity

We investigate the performance of machine learning algorithms trained exclusively with configurations obtained from importance sampling Monte Carlo simulations of the two-dimensional Ising model with conserved magnetization. For supervised machine learning, we use convolutional neural networks and find that the corresponding output not only allows to locate the phase transition point with high precision, it also displays a finite-size scaling characterized by an Ising critical exponent. For unsupervised learning, restricted Boltzmann machines (RBM) are trained to generate new configurations that are then used to compute various quantities. We find that RBM generates configurations with magnetizations and energies forbidden in the original physical system. The RBM generated configurations result in energy density probability distributions with incorrect weights as well as in wrong spatial correlations. We show that shortcomings are also encountered when training RBM with configurations obtained from the non-conserved Ising model.

Interpretable conservation law estimation by deriving the symmetries of dynamics from trained deep neural networks

Understanding complex systems with their reduced model is one of the central roles in scientific activities. Although physics has greatly been developed with the physical insights of physicists, it is sometimes challenging to build a reduced model of such complex systems on the basis of insight alone. We propose a framework that can infer hidden conservation laws of a complex system from deep neural networks (DNNs) that have been trained with physical data of the system. The purpose of the proposed framework is not to analyze physical data with deep learning but to extract interpretable physical information from trained DNNs. With Noether's theorem and by an efficient sampling method, the proposed framework infers conservation laws by extracting the symmetries of dynamics from trained DNNs. The proposed framework is developed by deriving the relationship between a manifold structure of a time-series data set and the necessary conditions for Noether's theorem. The feasibility of the proposed framework has been verified in some primitive cases in which the conservation law is well known. We also apply the proposed framework to conservation law estimation for a more practical case, that is, a large-scale collective motion system in the metastable state, and we obtain a result consistent with that of a previous study.

TensorFlow solver for quantum PageRank in large-scale networks

Google PageRank is a prevalent algorithm for ranking the significance of nodes or websites in a network, and a recent quantum counterpart for PageRank algorithm has been raised to suggest a higher accuracy of ranking comparing to Google PageRank. The quantum PageRank algorithm is essentially based on quantum stochastic walks and can be expressed using Lindblad master equation, which, however, needs to solve the Kronecker products of an O(N⁴) dimension and requires severely large memory and time when the number of nodes N in a network increases above 150. Here, we present an efficient solver for quantum PageRank by using the Runge-Kutta method to reduce the matrix dimension to O(N²) and employing TensorFlow to conduct GPU parallel computing. We demonstrate its performance in solving quantum stochastic walks on Erdös-Rényi graphs using an RTX 2060 GPU. The test on the graph of 6000 nodes requires a memory of 5.5 GB and time of 223 s, and that on the graph of 1000 nodes requires 226 MB and 3.6 s. Compared with QSWalk, a currently prevalent Mathematica solver, our solver for the same graph of 1000 nodes reduces the required memory and time to only 0.2% and 0.05%. We apply the solver to quantum PageRank for the USA major airline network with up to 922 nodes, and to quantum stochastic walk on a glued tree of 2186 nodes. This efficient solver for large-scale quantum PageRank and quantum stochastic walks would greatly facilitate studies of quantum information in real-life applications.

Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands

Multi-band insulating Bloch Hamiltonians with internal or spatial symmetries, such as particle-hole or inversion, may have topologically disconnected sectors of trivial atomic-limit (momentum-independent) Hamiltonians. We present a neural-network-based protocol for finding topologically relevant indices that are invariant under transformations between such trivial atomic-limit Hamiltonians, thus corresponding to the standard classification of band insulators. The work extends the method of ‘topological data augmentation’ for unsupervised learning introduced (2020 Phys. Rev. Res. 2 013354) by also generalizing and simplifying the data generation scheme and by introducing a special ‘mod’ layer of the neural network appropriate for Z n classification. Ensembles of training data are generated by deforming seed objects in a way that preserves a discrete representation of continuity. In order to focus the learning on the topologically relevant indices, prior to the deformation procedure we stack the seed Bloch Hamiltonians with a complete set of symmetry-respecting trivial atomic bands. The obtained datasets are then used for training an interpretable neural network specially designed to capture the topological properties by learning physically relevant momentum space quantities, even in crystalline symmetry classes.

Mapping distinct phase transitions to a neural network

We demonstrate, by means of a convolutional neural network, that the features learned in the two-dimensional Ising model are sufficiently universal to predict the structure of symmetry-breaking phase transitions in considered systems irrespective of the universality class, order, and the presence of discrete or continuous degrees of freedom. No prior knowledge about the existence of a phase transition is required in the target system and its entire parameter space can be scanned with multiple histogram reweighting to discover one. We establish our approach in q-state Potts models and perform a calculation for the critical coupling and the critical exponents of the ϕ^{4} scalar field theory using quantities derived from the neural network implementation. We view the machine learning algorithm as a mapping that associates each configuration across different systems to its corresponding phase and elaborate on implications for the discovery of unknown phase transitions.

Unsupervised Machine Learning of Quantum Phase Transitions Using Diffusion Maps

Experimental quantum simulators have become large and complex enough that discovering new physics from the huge amount of measurement data can be quite challenging, especially when little theoretical understanding of the simulated model is available. Unsupervised machine learning methods are particularly promising in overcoming this challenge. For the specific task of learning quantum phase transitions, unsupervised machine learning methods have primarily been developed for phase transitions characterized by simple order parameters, typically linear in the measured observables. However, such methods often fail for more complicated phase transitions, such as those involving incommensurate phases, valence-bond solids, topological order, and many-body localization. We show that the diffusion map method, which performs nonlinear dimensionality reduction and spectral clustering of the measurement data, has significant potential for learning such complex phase transitions unsupervised. This method may work for measurements of local observables in a single basis and is thus readily applicable to many experimental quantum simulators as a versatile tool for learning various quantum phases and phase transitions.

Machine learning for condensed matter physics

Condensed matter physics (CMP) seeks to understand the microscopic interactions of matter at the quantum and atomistic levels, and describes how these interactions result in both mesoscopic and macroscopic properties. CMP overlaps with many other important branches of science, such as chemistry, materials science, statistical physics, and high-performance computing. With the advancements in modern machine learning (ML) technology, a keen interest in applying these algorithms to further CMP research has created a compelling new area of research at the intersection of both fields. In this review, we aim to explore the main areas within CMP, which have successfully applied ML techniques to further research, such as the description and use of ML schemes for potential energy surfaces, the characterization of topological phases of matter in lattice systems, the prediction of phase transitions in off-lattice and atomistic simulations, the interpretation of ML theories with physics-inspired frameworks and the enhancement of simulation methods with ML algorithms. We also discuss in detail the main challenges and drawbacks of using ML methods on CMP problems, as well as some perspectives for future developments.

Unsupervised Phase Discovery with Deep Anomaly Detection

We demonstrate how to explore phase diagrams with automated and unsupervised machine learning to find regions of interest for possible new phases. In contrast to supervised learning, where data is classified using predetermined labels, we here perform anomaly detection, where the task is to differentiate a normal dataset, composed of one or several classes, from anomalous data. As a paradigmatic example, we explore the phase diagram of the extended Bose Hubbard model in one dimension at exact integer filling and employ deep neural networks to determine the entire phase diagram in a completely unsupervised and automated fashion. As input data for learning, we first use the entanglement spectra and central tensors derived from tensor-networks algorithms for ground-state computation and later we extend our method and use experimentally accessible data such as low-order correlation functions as inputs. Our method allows us to reveal a phase-separated region between supersolid and superfluid parts with unexpected properties, which appears in the system in addition to the standard superfluid, Mott insulator, Haldane-insulating, and density wave phases.

Topological Quantum Compiling with Reinforcement Learning

Quantum compiling, a process that decomposes the quantum algorithm into a series of hardware-compatible commands or elementary gates, is of fundamental importance for quantum computing. We introduce an efficient algorithm based on deep reinforcement learning that compiles an arbitrary single-qubit gate into a sequence of elementary gates from a finite universal set. It generates near-optimal gate sequences with given accuracy and is generally applicable to various scenarios, independent of the hardware-feasible universal set and free from using ancillary qubits. For concreteness, we apply this algorithm to the case of topological compiling of Fibonacci anyons and obtain near-optimal braiding sequences for arbitrary single-qubit unitaries. Our algorithm may carry over to other challenging quantum discrete problems, thus opening up a new avenue for intriguing applications of deep learning in quantum physics.

Extending machine learning classification capabilities with histogram reweighting

We propose the use of Monte Carlo histogram reweighting to extrapolate predictions of machine learning methods. In our approach, we treat the output from a convolutional neural network as an observable in a statistical system, enabling its extrapolation over continuous ranges in parameter space. We demonstrate our proposal using the phase transition in the two-dimensional Ising model. By interpreting the output of the neural network as an order parameter, we explore connections with known observables in the system and investigate its scaling behavior. A finite-size scaling analysis is conducted based on quantities derived from the neural network that yields accurate estimates for the critical exponents and the critical temperature. The method improves the prospects of acquiring precision measurements from machine learning in physical systems without an order parameter and those where direct sampling in regions of parameter space might not be possible.

Reducing autocorrelation times in lattice simulations with generative adversarial networks

Short autocorrelation times are essential for a reliable error assessment in Monte Carlo simulations of lattice systems. In many interesting scenarios, the decay of autocorrelations in the Markov chain is prohibitively slow. Generative samplers can provide statistically independent field configurations, thereby potentially ameliorating these issues. In this work, the applicability of neural samplers to this problem is investigated. Specifically, we work with a generative adversarial network (GAN). We propose to address difficulties regarding its statistical exactness through the implementation of an overrelaxation step, by searching the latent space of the trained generator network. This procedure can be incorporated into a standard Monte Carlo algorithm, which then permits a sensible assessment of ergodicity and balance based on consistency checks. Numerical results for real, scalar φ 4-theory in two dimensions are presented. We achieve a significant reduction of autocorrelations while accurately reproducing the correct statistics. We discuss possible improvements to the approach as well as potential solutions to persisting issues.

Machine learning identifies scale-free properties in disordered materials

The vast amount of design freedom in disordered systems expands the parameter space for signal processing. However, this large degree of freedom has hindered the deterministic design of disordered systems for target functionalities. Here, we employ a machine learning approach for predicting and designing wave-matter interactions in disordered structures, thereby identifying scale-free properties for waves. To abstract and map the features of wave behaviors and disordered structures, we develop disorder-to-localization and localization-to-disorder convolutional neural networks, each of which enables the instantaneous prediction of wave localization in disordered structures and the instantaneous generation of disordered structures from given localizations. We demonstrate that the structural properties of the network architectures lead to the identification of scale-free disordered structures having heavy-tailed distributions, thus achieving multiple orders of magnitude improvement in robustness to accidental defects. Our results verify the critical role of neural network structures in determining machine-learning-generated real-space structures and their defect immunity.

Machine learning as an improved estimator for magnetization curve and spin gap

The magnetization process is a very important probe to study magnetic materials, particularly in search of spin-liquid states in quantum spin systems. Regrettably, however, progress of the theoretical analysis has been unsatisfactory, mostly because it is hard to obtain sufficient numerical data to support the theory. Here we propose a machine-learning algorithm that produces the magnetization curve and the spin gap well out of poor numerical data. The plateau magnetization, its critical field and the critical exponent are estimated accurately. One of the hyperparameters identifies by its score whether the spin gap in the thermodynamic limit is zero or finite. After checking the validity for exactly solvable one-dimensional models we apply our algorithm to the kagome antiferromagnet. The magnetization curve that we obtain from the exact-diagonalization data with 36 spins is consistent with the DMRG results with 132 spins. We estimate the spin gap in the thermodynamic limit at a very small but finite value.

Short-sighted deep learning

A theory explaining how deep learning works is yet to be developed. Previous work suggests that deep learning performs a coarse graining, similar in spirit to the renormalization group (RG). This idea has been explored in the setting of a local (nearest-neighbor interactions) Ising spin lattice. We extend the discussion to the setting of a long-range spin lattice. Markov-chain Monte Carlo (MCMC) simulations determine both the critical temperature and scaling dimensions of the system. The model is used to train both a single restricted Boltzmann machine (RBM) network, as well as a stacked RBM network. Following earlier Ising model studies, the trained weights of a single-layer RBM network define a flow of lattice models. In contrast to results for nearest-neighbor Ising, the RBM flow for the long-ranged model does not converge to the correct values for the spin and energy scaling dimension. Further, correlation functions between visible and hidden nodes exhibit key differences between the stacked RBM and RG flows. The stacked RBM flow appears to move toward low temperatures, whereas the RG flow moves toward high temperature. This again differs from results obtained for nearest-neighbor Ising.

Systems Biology of Cancer Metastasis

Cancer metastasis is no longer viewed as a linear cascade of events but rather as a series of concurrent, partially overlapping processes, as successfully metastasizing cells assume new phenotypes while jettisoning older behaviors. The lack of a systemic understanding of this complex phenomenon has limited progress in developing treatments for metastatic disease. Because metastasis has traditionally been investigated in distinct physiological compartments, the integration of these complex and interlinked aspects remains a challenge for both systems-level experimental and computational modeling of metastasis. Here, we present some of the current perspectives on the complexity of cancer metastasis, the multiscale nature of its progression, and a systems-level view of the processes underlying the invasive spread of cancer cells. We also highlight the gaps in our current understanding of cancer metastasis as well as insights emerging from interdisciplinary systems biology approaches to understand this complex phenomenon.

Study of the Application of Deep Convolutional Neural Networks (CNNs) in Processing Sensor Data and Biomedical Images

The fast progress in research and development of multifunctional, distributed sensor networks has brought challenges in processing data from a large number of sensors. Using deep learning methods such as convolutional neural networks (CNN), it is possible to build smarter systems to forecasting future situations as well as precisely classify large amounts of data from sensors. Multi-sensor data from atmospheric pollutants measurements that involves five criteria, with the underlying analytic model unknown, need to be categorized, so do the Diabetic Retinopathy (DR) fundus images dataset. In this work, we created automatic classifiers based on a deep convolutional neural network (CNN) with two models, a simpler feedforward model with dual modules and an Inception Resnet v2 model, and various structural tweaks for classifying the data from the two tasks. For segregating multi-sensor data, we trained a deep CNN-based classifier on an image dataset extracted from the data by a novel image generating method. We created two deepened and one reductive feedforward network for DR phase classification. The validation accuracies and visualization results show that increasing deep CNN structure depth or kernels number in convolutional layers will not indefinitely improve the classification quality and that a more sophisticated model does not necessarily achieve higher performance when training datasets are quantitatively limited, while increasing training image resolution can induce higher classification accuracies for trained CNNs. The methodology aims at providing support for devising classification networks powering intelligent sensors.

Machine learning in electronic-quantum-matter imaging experiments

For centuries, the scientific discovery process has been based on systematic human observation and analysis of natural phenomena¹. Today, however, automated instrumentation and large-scale data acquisition are generating datasets of such large volume and complexity as to defy conventional scientific methodology. Radically different scientific approaches are needed, and machine learning (ML) shows great promise for research fields such as materials science^2-5. Given the success of ML in the analysis of synthetic data representing electronic quantum matter (EQM)^6-16, the next challenge is to apply this approach to experimental data-for example, to the arrays of complex electronic-structure images¹⁷ obtained from atomic-scale visualization of EQM. Here we report the development and training of a suite of artificial neural networks (ANNs) designed to recognize different types of order hidden in such EQM image arrays. These ANNs are used to analyse an archive of experimentally derived EQM image arrays from carrier-doped copper oxide Mott insulators. In these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unit-cell periodic, translational-symmetry-breaking EQM state. Further, the ANNs determine that this state is unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals^18,19 are consistent with these observations.

Machine learning topological defects of confined liquid crystals in two dimensions

Supervised machine learning can be used to classify images with spatially correlated physical features. We demonstrate the concept by using the coordinate files generated from an off-lattice computer simulation of rodlike molecules confined in a square box as an example. Because of the geometric frustrations at high number density, the nematic director field develops an inhomogeneous pattern containing various topological defects as the main physical feature. We describe two machine-learning procedures that can be used to effectively capture the correlation between the defect positions and the nematic directors around them and hence classify the topological defects. First is a feedforward neural network, which requires the aid of presorting the off-lattice simulation data in a coarse-grained fashion. Second is a recurrent neural network, which needs no such sorting and can be directly used for finding spatial correlations. The issues of when to presort a simulation data file and how the network structures affect such a decision are addressed.