# Machine Learning $\mathbb{Z}_{2}$ Quantum Spin Liquids with Quasi-particle Statistics

## TL;DR

This paper introduces a machine learning approach using quantum loop topography sensitive to quasi-particle statistics to efficiently map topological phase diagrams, specifically detecting a $ ext{Z}_2$ quantum spin liquid.

## Contribution

The authors develop a statistics-sensitive quantum loop topography method combined with neural networks to identify topological phases, improving efficiency over traditional techniques.

## Key findings

- Successfully identified the $ ext{Z}_2$ quantum spin liquid phase boundary.
- Demonstrated faster and more storage-efficient phase diagram evaluation.
- Showed the method's applicability to multi-parameter phase spaces.

## Abstract

After decades of progress and effort, obtaining a phase diagram for a strongly-correlated topological system still remains a challenge. Although in principle one could turn to Wilson loops and long-range entanglement, evaluating these non-local observables at many points in phase space can be prohibitively costly. With growing excitement over topological quantum computation comes the need for an efficient approach for obtaining topological phase diagrams. Here we turn to machine learning using quantum loop topography (QLT), a notion we have recently introduced. Specifically, we propose a construction of QLT that is sensitive to quasi-particle statistics. We then use mutual statistics between the spinons and visons to detect a $\mathbb{Z}_{2}$ quantum spin liquid in a multi-parameter phase space. We successfully obtain the quantum phase boundary between the topological and trivial phases using a simple feed-forward neural network. Furthermore, we demonstrate advantages of our approach for the evaluation of phase diagrams relating to speed and storage. Such statistics-based machine learning of topological phases opens new efficient routes to studying topological phase diagrams in strongly correlated systems.