Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks

Abstract

Correlator product states (CPS) are a powerful and very broad class of states for quantum lattice systems whose amplitudes can be sampled exactly and efficiently. They work by gluing together states of overlapping clusters of sites on the lattice, called correlators. Recently Carleo and Troyer Science 355, 602 (2017) introduced a new type sampleable ansatz called neural-network quantum states (NQS) that are inspired by the restricted Boltzmann model used in machine learning. By employing the formalism of tensor networks we show that NQS are a special form of CPS with novel properties. Diagramatically a number of simple observations become transparent. Namely, that NQS are CPS built from extensively sized GHZ-form correlators, which are related to a canonical polyadic decomposition of a tensor, making them uniquely unbiased geometrically. Another immediate implication of the equivalence to CPS is that we are able to formulate exact NQS representations for a wide range of paradigmatic states, including superposition of weighed-graph states, the Laughlin state, toric code states, and the resonating valence bond state. These examples reveal the potential of using higher dimensional hidden units and a second hidden layer in NQS. The major outlook of this study is the elevation of NQS to correlator operators allowing them to enhance conventional well-established variational Monte Carlo approaches for strongly correlated fermions.

40 pages, 27 figures. Comments most welcome

References in corpus (25)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Multi-party entanglement in graph states
• Learning phase transitions by confusion
• Criticality, the area law, and the computational power of PEPS
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• Self-Learning Monte Carlo Method
• Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• Unsupervised machine learning account of magnetic transitions in the Hubbard model
• Resonating valence bond states in the PEPS formalism
• An exact mapping between the Variational Renormalization Group and Deep Learning
• Strings, Projected Entangled Pair States, and variational Monte Carlo methods
• Chiral Topological Phases from Artificial Neural Networks
• Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules
• Ground state approximation for strongly interacting systems in arbitrary dimension
• Dynamical simulations of classical stochastic systems using matrix product states
• Deep learning and the renormalization group
• Variational ground states of 2D antiferromagnets in the valence bond basis
• Machine learning for many-body physics: efficient solution of dynamical mean-field theory
• Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices
• Variational Monte Carlo simulations using tensor-product projected states
• Capturing exponential variance using polynomial resources: applying tensor networks to non-equilibrium stochastic processes
• Measuring quantum entanglement, machine learning and wave function tomography: Bridging theory and experiment with the quantum gas microscope

Cited by in corpus (52)

• Machine learning and the physical sciences
• Reconstructing quantum states with generative models
• Symmetries and many-body excited states with neural-network quantum states
• Solving Many-Electron Schrödinger Equation Using Deep Neural Networks
• Backflow Transformations via Neural Networks for Quantum Many-Body Wave-Functions
• Constructing exact representations of quantum many-body systems with deep neural networks
• Machine Learning for Quantum Matter
• Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy
• Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
• Neural Gutzwiller-projected variational wave functions
• Chiral Topological Phases from Artificial Neural Networks
• The Variational Power of Quantum Circuit Tensor Networks
• Quenches near Ising quantum criticality as a challenge for artificial neural networks
• Self-learning Monte Carlo with Deep Neural Networks
• Efficient Representation of Topologically Ordered States with Restricted Boltzmann Machines
• Helping restricted Boltzmann machines with quantum-state representation by restoring symmetry
• Deep Learning-Enhanced Variational Monte Carlo Method for Quantum Many-Body Physics
• Generalized Transfer Matrix States from Artificial Neural Networks
• Tensor networks for unsupervised machine learning
• Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines
• Machine Learning Quantum States -- Extensions to Fermion-Boson Coupled Systems and Excited-State Calculations
• Correlation-Enhanced Neural Networks as Interpretable Variational Quantum States
• Real time evolution with neural-network quantum states
• Quantum Machine Learning Tensor Network States
• Restricted Boltzmann Machines and Matrix Product States of 1D Translational Invariant Stabilizer Codes
• Phase diagram of quantum generalized Potts-Hopfield neural networks
• Can neural quantum states learn volume-law ground states?
• Gaussian Process States: A data-driven representation of quantum many-body physics
• Unifying view of fermionic neural network quantum states: From neural network backflow to hidden fermion determinant states
• Spin-1/2 kagome Heisenberg antiferromagnet: Machine learning discovery of the spinon pair density wave ground state
• The quantum Gaussian process state: A kernel-inspired state with quantum support data
• On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension
• Sampling scheme for neuromorphic simulation of entangled quantum systems
• Compact Neural-network Quantum State representations of Jastrow and Stabilizer states
• Impact of conditional modelling for a universal autoregressive quantum state
• Efficient neural-network based variational Monte Carlo scheme for direct optimization of excited energy states in frustrated quantum systems
• Measures of distinguishability between stochastic processes
• Neural-network Quantum States for Spin-1 systems: spin-basis and parameterization effects on compactness of representations
• Classical Quantum Optimization with Neural Network Quantum States
• Investigating Network Parameters in Neural-Network Quantum States
• Exact block encoding of imaginary time evolution with universal quantum neural networks
• Specialising Neural-network Quantum States for the Bose Hubbard Model
• Boltzmann machines and quantum many-body problems
• Machine Learning Wavefunction
• Efficiency of the hidden fermion determinant states Ansatz in the light of different complexity measures
• Entanglement-informed Construction of Variational Quantum Circuits
• Efficiency of neural-network state representations of one-dimensional quantum spin systems
• Seeding neural network quantum states with tensor network states
• Emergence of global receptive fields capturing multipartite quantum correlations
• Neural Tangent Kernel of Matrix Product States: Convergence and Applications
• Basis dependence of Neural Quantum States for the Transverse Field Ising Model
• Brute-force positivization of $J_1-J_2$ model ground states