Analyzing quantum machine learning using tensor network

Variational quantum machine learning (VQML), which employs variational quantum circuits as computational models for machine learning, is considered one of the most promising applications for near-term quantum devices. We represent a VQML model as a tensor network (TN) and analyze it in the context of the TN. We identify the model as a featured linear model (FLM) with a constrained coefficient where the feature map is given by the tensor products. This allows us to create the same feature map classically in an efficient way using only the same amount of pre-processing as VQML, resulting in a classical TN machine learning model that exists within the function space spanned by the same basis functions as VQML models. By representing the coefficient components of the models using matrix product states (MPS), we analyze the coefficients of the VQML model and determine the conditions for efficient approximation of VQML models by classical models. Finally, we compare the performance of the VQML and classical models in function regression tasks using kernel and variational methods, highlighting the distinct characteristics between them. Our work presents a consolidated approach to comparing classical and quantum machine learning models within the unified framework of tensor network.
Comment: 18 + 6 pages, 12 + 5 figures