# Unsupervised machine learning of quantum phase transitions using diffusion maps

arXiv: https://arxiv.org/abs/2003.07399 · HTML: https://mlsanity.com/paper/2003.07399/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2020.

## Built on by (most cited first, 5 of 5)

- [Exponentially improved efficient machine learning for quantum many-body states with provable guarantees](https://mlsanity.com/paper/2304.04353/) (2023)
- [Quantum diffusion map for nonlinear dimensionality reduction](https://mlsanity.com/paper/2106.07302/) (2021)
- [Snake net and balloon force with a neural network for detecting multiple phases](https://mlsanity.com/paper/2205.09699/) (2022)
- [Attention to Order: Transformers Discover Phase Transitions via Learnability](https://mlsanity.com/paper/2510.07401/) (2025)
- [TetrisCNN for interpretable detection of phases of matter from experimental quantum simulator data](https://mlsanity.com/paper/2609.20693/) (2026)

## Abstract (arXiv)

Experimental quantum simulators have become large and complex enough that discovering new physics from the huge amount of measurement data can be quite challenging, especially when little theoretical understanding of the simulated model is available. Unsupervised machine learning methods are particularly promising in overcoming this challenge. For the specific task of learning quantum phase transitions, unsupervised machine learning methods have primarily been developed for phase transitions characterized by simple order parameters, typically linear in the measured observables. However, such methods often fail for more complicated phase transitions, such as those involving incommensurate phases, valence-bond solids, topological order, and many-body localization. We show that the diffusion map method, which performs nonlinear dimensionality reduction and spectral clustering of the measurement data, has significant potential for learning such complex phase transitions unsupervised. This method works for measurements of local observables in a single basis and is thus readily applicable to many experimental quantum simulators as a versatile tool for learning various quantum phases and phase transitions.