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arXiv:1710.04713 · Physical review. B./Physical review. B

Chiral topological phases from artificial neural networks

Raphael Kaubruegger, Lorenzo Pastori, Jan Carl Budich

Published May 18, 2018• 89 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.97.195136

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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.
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.
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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.
Quantum matter, the research field studying phases of matter whose properties are intrinsically quantum mechanical, draws from areas as diverse as hard condensed matter physics, materials science, statistical mechanics, quantum information, quantum gravity, and large-scale numerical simulations. Recently, researchers interested in quantum matter and strongly correlated quantum systems have turned their attention to the algorithms underlying modern machine learning with an eye on making progress in their fields. Here we provide a short review on the recent development and adaptation of machine learning ideas for the purpose advancing research in quantum matter, including ideas ranging from algorithms that recognize conventional and topological states of matter in synthetic experimental data, to representations of quantum states in terms of neural networks and their applications to the simulation and control of quantum systems. We discuss the outlook for future developments in areas at the intersection between machine learning and quantum many-body physics.
Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.