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arXiv:1609.09060 · Physical review. B./Physical review. B

Machine learning topological states

Dong-Ling Deng, Xiaopeng Li, S. Das Sarma

Published Nov 22, 2017• 289 citations
arXivHTMLPDF

Abstract

Machine learning, the core of artificial intelligence and data science, is a very active field, with vast applications throughout science and technology. Recently, machine learning techniques have been adopted to tackle intricate quantum many-body problems and phase transitions. In this work, the authors construct exact mappings from exotic quantum states to machine learning network models. This work shows for the first time that the restricted Boltzmann machine can be used to study both symmetry-protected topological phases and intrinsic topological order. The exact results are expected to provide a substantial boost to the field of machine learning of phases of matter.

DOI
10.1103/physrevb.96.195145

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Machine learning, the core of artificial intelligence and data science, is a very active field, with vast applications throughout science and technology. Recently, machine learning techniques have been adopted to tackle intricate quantum many-body problems and phase transitions. In this work, the authors construct exact mappings from exotic quantum states to machine learning network models. This work shows for the first time that the restricted Boltzmann machine can be used to study both symmetry-protected topological phases and intrinsic topological order. The exact results are expected to provide a substantial boost to the field of machine learning of phases of matter.
Artificial neural networks and machine learning tools have become ubiquitous far beyond the field of computer science. They already smartly assist us in our daily lives in various ways. Recently, these concepts have been adopted and applied in the realm of quantum physics, for example as a computational Ansatz for the state of a quantum many-body system. In this work, the authors harness the enormous flexibility of artificial neural networks to study exotic phases of quantum matter, known as chiral topological phases, that are particularly hard to investigate microscopically with more conventional computational methods.
Artificial neural networks play a prominent role in the rapidly growing field of machine learning and are recently introduced to quantum many-body systems to tackle complex problems. Here, we find that even topological states with long-range quantum entanglement can be represented with classical artificial neural networks. This is demonstrated by using two concrete spin systems, the one-dimensional (1D) symmetry-protected topological cluster state and the 2D toric code state with an intrinsic topological order. For both cases we show rigorously that the topological ground states can be represented by short-range neural networks in an {\it exact} fashion. This neural network representation, in addition to being exact, is surprisingly {\it efficient} as the required number of hidden neurons is as small as the number of physical spins. Our exact construction of topological-order neuron-representation demonstrates explicitly the exceptional power of neural networks in describing exotic quantum states, and at the same time provides valuable topological data to supervise machine learning topological quantum orders in generic lattice models.
The application of state-of-the-art machine learning techniques to statistical physic problems has seen a surge of interest for their ability to discriminate phases of matter by extracting essential features in the many-body wavefunction or the ensemble of correlators sampled in Monte Carlo simulations. Here we introduce a gener- alization of supervised machine learning approaches that allows to accurately map out phase diagrams of inter- acting many-body systems without any prior knowledge, e.g. of their general topology or the number of distinct phases. To substantiate the versatility of this approach, which combines convolutional neural networks with quantum Monte Carlo sampling, we map out the phase diagrams of interacting boson and fermion models both at zero and finite temperatures and show that first-order, second-order, and Kosterlitz-Thouless phase transitions can all be identified. We explicitly demonstrate that our approach is capable of identifying the phase transition to non-trivial many-body phases such as superfluids or topologically ordered phases without supervision.
We report an experimental demonstration of a machine learning approach to identify exotic topological phases, with a focus on the three-dimensional chiral topological insulators. We show that the convolutional neural networks-a class of deep feed-forward artificial neural networks with widespread applications in machine learning-can be trained to successfully identify different topological phases protected by chiral symmetry from experimental raw data generated with a solid-state quantum simulator. Our results explicitly showcase the exceptional power of machine learning in the experimental detection of topological phases, which paves a way to study rich topological phenomena with the machine learning toolbox.
Understanding the phases of a model usually requires knowledge of their characteristic features, which are nonlocal in topologically ordered systems. Here, the authors reframe the phase classification problem in disordered topological superconductors as a data-driven task, motivated by the recent surge of interest in the application of machine-learning techniques including deep learning. It is demonstrated that an artificial neural network learns to extract the essence of the clean system and successfully distinguishes phases even under disorder by statistical recovery of translational symmetry.
We introduce a machine learning framework for extracting Ising topological order from defect partition functions of the two-dimensional Ising model on a torus. Restricted Boltzmann Machines (RBMs) are trained on Ising model data sampled at criticality across topological sectors. We take a component-wise square-root map of the learned distributions which naturally produces candidate wavefunctions for the (2+1)-dimensional Ising TQFT. As a nontrivial consistency check, we extract the modular S-matrix from overlaps of the resulting states and recover the expected Ising modular data. Our results demonstrate that neural network representations can capture both critical fluctuations and emergent topological structure, providing a data-driven route from lattice statistical mechanics to topological quantum field theory.