Quaternion-based machine learning on topological quantum systems

Abstract

Topological phase classifications have been intensively studied via machine-learning techniques where different forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the complexity in 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 and 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 a conventional convolutional neural network. The machine takes quaternion-transformed configurations as inputs and successfully classify all distinct topological phases, even for those states that have different distributuions from those states seen by the
machine during the training process. Our work demonstrates the power of quaternion algebras on extracting crucial features from the targeted data and the advantages of quaternion-based neural networks than conventional ones in the tasks of topological phase classifications.

Multi-scale quaternion CNN and BiGRU with cross self-attention feature fusion for fault diagnosis of bearing

“…Based on the particular form of quaternion, it has advantages in representing and processing more complex data and obtaining rich features. Therefore, quaternion neural network have been successfully applied in many fields in recent years, such as semantic segmentation method [42], endoscopic clinical diagnosis [43], topological phase classification [44], and random noise suppression of seismic data [45]. In the above works, quaternion are the expansion of complex numbers in four-dimensional space, with one fundamental part and three imaginary parts, which can represent the spatial rotation with more flexibility.…”