Gaussian Process States: A data-driven representation of quantum many-body physics

Abstract

We present a novel, non-parametric form for compactly representing entangled many-body quantum states, which we call a `Gaussian Process State'. In contrast to other approaches, we define this state explicitly in terms of a configurational data set, with the probability amplitudes statistically inferred from this data according to Bayesian statistics. In this way the non-local physical correlated features of the state can be analytically resummed, allowing for exponential complexity to underpin the ansatz, but efficiently represented in a small data set. The state is found to be highly compact, systematically improvable and efficient to sample, representing a large number of known variational states within its span. It is also proven to be a `universal approximator' for quantum states, able to capture any entangled many-body state with increasing data set size. We develop two numerical approaches which can learn this form directly: a fragmentation approach, and direct variational optimization, and apply these schemes to the Fermionic Hubbard model. We find competitive or superior descriptions of correlated quantum problems compared to existing state-of-the-art variational ansatzes, as well as other numerical methods.

References in corpus (41)

• The density-matrix renormalization group in the age of matrix product states
• Three qubits can be entangled in two inequivalent ways
• Gaussian Approximation Potentials: the accuracy of quantum mechanics, without the electrons
• Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Stripe order in the underdoped region of the two-dimensional Hubbard model
• Deep Neural Networks as Gaussian Processes
• Deep neural network solution of the electronic Schrödinger equation
• Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks
• Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms
• Generalized Lanczos Algorithm for Variational Quantum Monte Carlo
• Efficient Representation of Quantum Many-body States with Deep Neural Networks
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• A spin-adapted Density Matrix Renormalization Group algorithm for quantum chemistry
• Recurrent Neural Network Wave Functions
• Fermionic neural-network states for ab-initio electronic structure
• Accurate Interatomic Force Fields via Machine Learning with Covariant Kernels
• Quantum Entanglement in Deep Learning Architectures
• Ground-state phase diagram of the square lattice Hubbard model from density matrix embedding theory
• Additive Gaussian Processes
• Backflow Transformations via Neural Networks for Quantum Many-Body Wave-Functions
• Variational quantum Monte Carlo simulations with tensor-network states
• Constructing exact representations of quantum many-body systems with deep neural networks
• Variational Monte Carlo Method Combined with Quantum-Number Projection and Multi-Variable Optimization
• Neural Quantum States of frustrated magnets: generalization and sign structure
• Approximating strongly correlated spin and fermion wavefunctions with correlator product states
• Solving frustrated quantum many-particle models with convolutional neural networks
• Ground-State Properties of Quantum Many-Body Systems: Entangled-Plaquette States and Variational Monte Carlo
• Quantum Neural Network States: A Brief Review of Methods and Applications
• Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method
• mVMC - Open-source software for many-variable variational Monte Carlo method
• Neural network wave functions and the sign problem
• Phase separation in the Hubbard model
• Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks
• Quantum-number projection in the path-integral renormalization group method
• Transfer learning for scalability of neural-network quantum states
• Absence of static stripes in the two-dimensional $t-J$ model by an accurate and systematic quantum Monte Carlo approach
• Lattice effects on Laughlin wave functions and parent Hamiltonians
• Iterative Retraining of Quantum Spin Models Using Recurrent Neural Networks
• A Bayesian Inference Framework for Compression and Prediction of Quantum States

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