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arXiv:2301.02683 · Physical review. B./Physical review. B

Clustering neural quantum states via diffusion maps

Yanting Teng, Subir Sachdev A, Mathias S. Scheurer

Published Nov 27, 2023• 3 citations
arXivHTMLPDF

Abstract

We discuss and demonstrate an unsupervised machine-learning procedure to detect topological order in quantum many-body systems. Using a restricted Boltzmann machine to define a variational Ansatz for the low-energy spectrum, we sample wave functions with probability decaying exponentially with their variational energy; this defines our training dataset that we use as input to a diffusion map scheme. The diffusion map provides a low-dimensional embedding of the wave functions, revealing the presence or absence of superselection sectors and, thus, topological order. We show that for the diffusion map, the required similarity measure of quantum states can be defined in terms of the network parameters, allowing for an efficient evaluation within polynomial time. However, possible ``gauge redundancies'' have to be carefully taken into account. As an explicit example, we apply the method to the toric code.

DOI
10.1103/physrevb.108.205152

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We discuss and demonstrate an unsupervised machine-learning procedure to detect topological order in quantum many-body systems. Using a restricted Boltzmann machine to define a variational Ansatz for the low-energy spectrum, we sample wave functions with probability decaying exponentially with their variational energy; this defines our training dataset that we use as input to a diffusion map scheme. The diffusion map provides a low-dimensional embedding of the wave functions, revealing the presence or absence of superselection sectors and, thus, topological order. We show that for the diffusion map, the required similarity measure of quantum states can be defined in terms of the network parameters, allowing for an efficient evaluation within polynomial time. However, possible ``gauge redundancies'' have to be carefully taken into account. As an explicit example, we apply the method to the toric code.
Despite their fundamental importance in dictating the quantum-mechanical properties of a system, ground states of many-body local quantum Hamiltonians form a set of measure zero in the many-body Hilbert space. Hence determining whether a given many-body quantum state is ground-stateable is a challenging task. Here we propose an unsupervised machine learning approach, dubbed Entanglement Clustering ("EntanCl"), to separate out ground-stateable wave functions from those that must be excited-state wave functions using entanglement structure information. EntanCl uses snapshots of an ensemble of swap operators as input and projects these high-dimensional data to two dimensions, preserving important topological features of the data associated with distinct entanglement structure using the uniform manifold approximation and projection. The projected data are then clustered using K-means clustering with k = 2. By applying EntanCl to two examples, a one-dimensional free fermion model and the two-dimensional toric code, we demonstrate that EntanCl can successfully separate ground states from excited states with high computational efficiency. Being independent of a Hamiltonian and associated energy estimates, EntanCl offers a new paradigm for addressing quantum many-body wave functions in a computationally efficient manner.
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.
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.
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.
Topological invariants allow one to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wave functions under twisted boundary conditions. However, those procedures do not allow one to calculate a topological invariant by evaluating the system locally, and thus require information about the wave functions in the whole system. Here we show that artificial neural networks can be trained to identify the topological order by evaluating a local projection of the density matrix. We demonstrate this for two different models, a one-dimensional topological superconductor and a two-dimensional quantum anomalous Hall state, both with spatially modulated parameters. Our neural network correctly identifies the different topological domains in real space, predicting the location of in-gap states. By combining a neural network with a calculation of the electronic states that uses the kernel polynomial method, we show that the local evaluation of the invariant can be carried out by evaluating a local quantity, in particular for systems without translational symmetry consisting of tens of thousands of atoms. Our results show that supervised learning is an efficient methodology to characterize the local topology of a system.
Despite rapidly growing interest in harnessing machine learning in the study of quantum many-body systems, training neural networks to identify quantum phases is a nontrivial challenge.The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network.When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of nonlocal properties.Here, we introduce quantum loop topography (QLT): a procedure of constructing a multidimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps.The loop configuration is guided by the characteristic response for defining the phase, which is Hall conductivity for the cases at hand.Feeding QLT to a fully connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish the Chern insulator and the fractional Chern insulator from trivial insulators with high fidelity.In addition to establishing the first case of obtaining a phase diagram with a topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.