Incorporation of quaternion algebras in machine learning for topological quantum systems

Min-Ruei Lin¹, Wan-Ju Li¹, Shin-Ming Huang^1,2*

¹Department of Physics, National Sun Yat-sen University, Kaohsiung, Taiwan
²Center of Crystal Research, National Sun Yat-sen University, Kaohsiung, Taiwan

* Presenter:Shin-Ming Huang, email:shinming@mail.nsysu.edu.tw

Topological phase classifications have been intensively studied via machine-learning techniques
where different forms of training data are proposed in order to maximize the information extracted from the systems of interest. Due to the complexity of quantum physics, advanced mathematical architecture should be considered in designing machines. In this work, we incorporate
quaternion algebras into data analysis either in the frame of supervised or unsupervised learning to classify two-dimensional Chern insulators. For the unsupervised-learning aspect, we apply
the principal component analysis (PCA) on the quaternion-transformed eigenstates to distinguish
topological phases. For the supervised-learning aspect, we construct our machine by adding one
quaternion convolutional layer on top of conventional convolutional neural networks. The machine takes quaternion-transformed configurations as inputs and successfully classifies all distinct
topological phases, even for those structures that are not seen by the machine during the training
process. Our work demonstrates the power of quaternion algebras in extracting crucial features
from the targeted data and the advantages of quaternion-based neural networks over conventional
ones in the tasks of topological phase classifications.

Keywords: quaternion, machine learning, convolutional neural network, Chern insulator, principal component analysis