Session E18: Machine Learning Quantum States I

Learning quantum states with generative models

The technological success of machine learning techniques has motivated a research area in the condensed matter physics and quantum information communities, where new tools and conceptual connections between machine learning and many-body physics are rapidly developing. In this talk, I will discuss the use of generative models for learning quantum states. In particular, I will discuss a strategy for learning mixed states through a combination of informationally complete positive-operator valued measures and generative models. In this setting, generative models enable accurate learning of prototypical quantum states of large size directly from measurements mimicking experimental data.   

Machine Learning Holography in Neural Network Renormalization Group

Previously, people have shown the close relations between renormalization group(RG) with both deep learning and holographic duality, and how holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. Inspired by those, we propose any boundary conformal field theory can be mapped into its holographic bulk to an operator level. In our framework, renormalization group is constructed as a hierarchical unsupervised generative model. Coarse graining direction can be viewed as an emergent direction, and it pushes boundary field theory configurations to bulk field configurations. The inverse coarse graining direction generates boundary field configurations from bulk noises. The goal is to construct optimal RG that makes bulk variables as uncorrelated as possible. The leftover of correlations between bulk variables can be used to define measure of distance in the bulk. We studied two dimensional interacting bosonic system as a boundary field theory. RG network is trained to find the effective bulk field theory and we observed the emergence of hyperbolic geometry(AdS3 spatial geometry) as we tuned system towards critical point.   

Analytic continuation by combining sparse modeling with the Pade approximation

Numerical methods based on the imaginary-time path-integral such as the path-integral Monte Carlo method are powerful tools to investigate a quantum many-body system both at absolute zero and finite temperatures. For example, these can calculate the imaginary-time Green's function directly, and other important quantities such as spectrum function can be transformed from this by the analytic continuation (AC) or by solving the Lehmann representation as an integral equation. In practice, however, this transformation is unstable against noise of the imaginary-time Green's function.

Several methods have been developed to solve this problem so far. The noise reduction by the sparse modeling (SpM) is one of them. In this method, we transform basis by the singular matrices of the integral kernel and truncate noisy components in the new basis by the sparse modeling. However, this truncation introduces an unphysical oscillation to the obtained spectrum as a systematic error.

In this study, we have improved SpM method by combining it with AC by the Pade approximation. AC by the Pade approximation gives a stable and smooth spectrum in low frequency region, which can be used to make the SpM spectrum smooth and high accuracy.   

A machine learning approach to excited states of quantum many-body systems

We present a variational Monte Carlo method for determining dynamical properties and the spectral function of quantum many-body systems. Restricted Boltzmann machines (RBMs) are used to encode the Green's function of the system. First, the ground state wave function is calculated using a standard variational approach. The dynamical correlation function is then obtained by solving two linear systems of equations. We present a variational Monte Carlo approach to do so. This process has to be repeated for each value of frequency and momentum, but it can be easily parallelized. We illustrate it with applications to the Heisenberg model in one and two dimensions. Results show remarkable agreement with exact calculations on small systems and demonstrate that RBMs can also faithfully represent excited states.   

Machine Learning Spatial Geometry from Entanglement Features

Motivated by the close relations of the renormalization group with both the holography duality and the deep learning, we propose that the holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. We develop a concrete algorithm, call the entanglement feature learning (EFL), based on the random tensor network (RTN) model for the tensor network holography. We show that each RTN can be mapped to a Boltzmann machine, trained by the entanglement entropies over all subregions of a given quantum many-body state. The goal is to construct the optimal RTN that best reproduce the entanglement feature. The RTN geometry can then be interpreted as the emergent holographic geometry. We demonstrate the EFL algorithm on 1D free fermion system and observe the emergence of the hyperbolic geometry (AdS3 spatial geometry) as we tune the fermion system towards the gapless critical point (CFT2 point).   

Machine learning many-body localization: Search for the elusive nonergodic metal

The many-body localization transition in isolated quantum systems with a single-particle mobility edge is an intriguing subject that has not yet been fully understood. In particular, whether a nonergodic metallic phase associated with a many-body mobility edge exists or not is under active debate. In this Letter, we construct a neural network classifier to investigate the existence of the nonergodic metallic phase in a prototype model using many-body entanglement spectra as the sole diagnostic. We find that such a classifier is able to identify with high confidence the nonergodic metallic phase existing between the many-body localized and the thermal phase. Our neural network based approach shows how supervised machine learning can be applied not only in locating phase boundaries, but also in providing a way to definitively examine the existence of a novel phase whose existence is unclear.   

Machine learning of condensed-matter phases with physical interpretability

Nature displays a vast variety of phase transitions. Detecting and quantifying these often requires creative design of order parameters, which are strongly system dependent. Thanks to recent advances in machine learning (ML), it is now possible to identify phase behavior by directly analyzing atomistic configurations via modern pattern-recognition techniques. Whereas these ML approaches are universal and robust, they often suffer from the lack of physical interpretability. Here we introduce a new ML scheme, which distinguishes different condensed-matter phases by autonomously recognizing order parameters. When applied to two-dimensional Ising models, our method can accurately predict the Curie and Néel temperatures and capture the corresponding critical phenomena by recognizing ferro- and antiferro-magnetizations, respectively. Going beyond these prototypical test cases, we analyze the nonequilibrium polymeric sol–gel transition, locating not only the transition temperature, but also discovering two classes of underlying collective behavior, the condensation and network-formation modes. Compared to existing MLs, our method offers physical insights as well as high training efficiency.
  

Interpretable Machine Learning Study of Many-Body Localization Transition in Disordered Quantum Spin Chains

We develop, train, and apply a support vector machine (SVM) to study the phase transition between many-body localized and thermal phases in a disordered quantum Ising chain. We use the labeled probability density of eigenstate wavefunctions in the deeply localized and thermal regimes at two different energy densities as the training set. We find that the trained SVM is then able to predict the whole phase diagram. The obtained phase boundary qualitatively agrees with previous work using entanglement entropy to characterize these two phases. We further analyze the decision function of the SVM to interpret its physical meaning and find that it is analogous to the inverse participation ratio in the many-body configuration space. Our findings demonstrate the ability of the SVM to capture potential quantities that may characterize the many-body localization phase transition. The qualitative agreement of phase boundary obtained by SVM and by scaling entanglement entropy motivates further exploration of the relation between these two different quantities in connection to many-body localization.   

Analytic continuation via “domain-knowledge free” machine learning

We present a machine-learning (ML) approach to a long-standing issue in quantum many-body physics, namely, analytic continuation. This notorious ill-conditioned problem of obtaining spectral function from Green’s function has been a focus of new method developments for past decades. Here we demonstrate the usefulness of modern ML techniques including convolutional neural networks and the variants of stochastic gradient descent optimizer. ML continuation kernel is successfully realized without any ‘domain-knowledge’, which means that any physical ‘prior’ is not utilized in the kernel construction and the neural networks ‘learn’ the knowledge solely from ‘training’. The outstanding performance is achieved for both insulating and metallic band structure. Our ML-based approach not only provides the more accurate spectrum than the conventional methods in terms of peak positions and heights, but is also more robust against the noise which is the required key feature for any continuation technique to be successful [1]. Furthermore, its computation speed is 10⁴–10⁵ times faster than maximum entropy method.

[1] H. Yoon, J.-H. Sim, and M. J. Han, ArXiv:1806.03841.   

Monte Carlo Renormalization Group for Systems with Quenched Disorder

We extend to quenched disordered systems the variational scheme for real space renormalization
group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder
is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from
which one can compute the critical exponents if the correlations of the renormalized couplings retain
finite range. Key to the variational approach is the bias potential found by minimizing a convex
functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation
time in large disordered systems. We demonstrate the method with applications to dilute Ising,
random field Ising and short-range spin glass models, on the two-dimensional square lattice.