Quantum-Classical Machine learning by Hybrid Tensor Networks

TL;DR

The paper addresses the limitations of regular tensor networks as deep-learning blocks, notably restricted representation power and poor scalability, by introducing Hybrid Tensor Networks (HTN) that fuse TN layers with classical neural networks in a unified deep-learning framework trainable via backpropagation and SGD. HTN enables nonlinear learning and supports both entangled and product quantum states, demonstrated through HTN architectures for quantum-state classification and a quantum-classical autoencoder, with practical MNIST and Fashion-MNIST experiments. The work further explores parameterized quantum circuits as a means to simulate HTN on quantum hardware, reporting promising results on a $4\times4$ MNIST subset for classification and a $2\times2$ reconstruction task with PSNR $=22.72$. Overall, HTN offers a flexible, scalable path toward hybrid quantum-classical deep learning and quantum feature engineering, with potential extensions to entangled-state data and PQC-based implementations.

Abstract

Tensor networks (TN) have found a wide use in machine learning, and in particular, TN and deep learning bear striking similarities. In this work, we propose the quantum-classical hybrid tensor networks (HTN) which combine tensor networks with classical neural networks in a uniform deep learning framework to overcome the limitations of regular tensor networks in machine learning. We first analyze the limitations of regular tensor networks in the applications of machine learning involving the representation power and architecture scalability. We conclude that in fact the regular tensor networks are not competent to be the basic building blocks of deep learning. Then, we discuss the performance of HTN which overcome all the deficiency of regular tensor networks for machine learning. In this sense, we are able to train HTN in the deep learning way which is the standard combination of algorithms such as Back Propagation and Stochastic Gradient Descent. We finally provide two applicable cases to show the potential applications of HTN, including quantum states classification and quantum-classical autoencoder. These cases also demonstrate the great potentiality to design various HTN in deep learning way.

Figures (10)

• Figure 1: Universal framework of Hybrid Tensor Networks.
• Figure 2: Different ways of message passing on Neural Networks and Tensor Networks. The directions of message passing are denoted by blue arrows. For tensor network, the message passing is implemented by the operation of tensor contraction; (a) Neural Network ; (b) Regular tensor network; (c) Generalized tensor network, and the operation of copy is marked by red dot;
• Figure 3: Speeding up on triangle tensor network contraction by GPU platform. The time cost is plotted on logarithmic y-axis.
• Figure 4: Training loss of quantum-classical autoencoder; (a) MNIST ; (b) Fashion-MNIST ;
• Figure 5: HTN for quantum states classification. We embed two tree tensor network layers and three dense neural network layers.