Constructing exact representations of quantum many-body systems with deep neural networks

TL;DR

This paper introduces a deterministic deep neural network method using deep Boltzmann machines to exactly represent ground states of many-body quantum systems, avoiding stochastic optimization and enabling efficient sampling.

Contribution

It presents a novel, constructive approach to generate exact neural network representations of quantum ground states without stochastic training, applicable to large systems.

Findings

Linear growth of neurons with system size and imaginary time

Exact representation of ground states for specific models

Efficient sampling of physical and neuron configurations

Abstract

We develop a constructive approach to generate artificial neural networks representing the exact ground states of a large class of many-body lattice Hamiltonians. It is based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations among physical degrees of freedom in the visible layer. The approach reproduces the exact imaginary-time Hamiltonian evolution, and is completely deterministic. In turn, compact and exact network representations for the ground states are obtained without stochastic optimization of the network parameters. The number of neurons grows linearly with the system size and total imaginary time, respectively. Physical quantities can be measured by sampling configurations of both physical and neuron degrees of freedom. We provide specific examples for the transverse-field Ising and Heisenberg models by implementing efficient sampling. As a compact, classical representation for many-body quantum systems, our approach is an alternative to the standard path integral, and it is potentially useful also to systematically improve on numerical approaches based on the restricted Boltzmann machine architecture.