Classifying topological neural network quantum states via diffusion maps

TL;DR

This paper introduces an unsupervised machine learning approach using diffusion maps and restricted Boltzmann machines to detect topological order in quantum many-body systems efficiently, demonstrated on the toric code.

Contribution

The paper presents a novel method combining diffusion maps and neural network wave functions to identify topological phases without supervision.

Findings

Successfully applied to the toric code model.

Efficient polynomial-time evaluation of quantum state similarities.

Revealed topological order through low-dimensional embedding.

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.