Unsupervised learning of quantum phase transitions using nonlinear dimension reduction methods

Quantum simulators have reached a complexity that understanding measurement data they generated has become a daunting task for traditional data analysis methods. To make scientific discoveries based on experimental data where theoretical understanding is lacking, unsupervised machine learning can be a powerful tool. However, existing approaches to unsupervised learning of quantum many-body states are largely focused linear dimension reduction methods such as principle component analysis (PCA), or generalizations such as kernel PCA. These methods often fail when order parameters of the states are nonlinear functions, i.e. states with valence-bond order, topological order, many-body localization, etc. This motivates us to investigate nonlinear dimension reduction methods such as diffusion maps and autoencoders. By studying a 1D chiral Z3 clock model (experimentally a chain of Rydberg atoms), we find PCA detects only the Z3 phase while diffusion maps detects the full phase map including a incommensurate phase. In addition, diffusion maps directly detect the Z3 symmetry of model and predict the number of clusters. We find these nonlinear dimension reduction methods also useful in learning valence-bond order and Gaussian-type topological phase transitions in quantum spin systems.