Tensor networks (TNs) and neural networks (NNs) are two fundamental data modeling approaches. TNs were introduced to solve the curse of dimensionality in large-scale tensors by converting an exponential number of dimensions to polynomial complexity. As a result, they have attracted significant attention in the fields of quantum physics and machine learning. Meanwhile, NNs have displayed exceptional performance in various applications, e.g., computer vision, natural language processing, and robotics research. Interestingly, although these two types of networks originate from different observations, they are inherently linked through the common multilinearity structure underlying both TNs and NNs, thereby motivating a significant number of intellectual developments regarding combinations of TNs and NNs. In this paper, we refer to these combinations as tensorial neural networks (TNNs), and present an introduction to TNNs in three primary aspects: network compression, information fusion, and quantum circuit simulation. Furthermore, this survey also explores methods for improving TNNs, examines flexible toolboxes for implementing TNNs, and documents TNN development while highlighting potential future directions. To the best of our knowledge, this is the first comprehensive survey that bridges the connections among NNs, TNs, and quantum circuits. We provide a curated list of TNNs at \url{https://github.com/tnbar/awesome-tensorial-neural-networks}.

翻译：张量网络（TNs）和神经网络（NNs）是两种基础的数据建模方法。张量网络通过将指数级维度转换为多项式复杂度，解决了大规模张量中的维度灾难问题，因此在量子物理和机器学习领域引起了广泛关注。与此同时，神经网络在计算机视觉、自然语言处理和机器人研究等各类应用中表现出卓越性能。有趣的是，尽管这两种网络源自不同的观察角度，但它们通过张量网络和神经网络中共同的多元线性结构天然地相互关联，从而催生了大量关于二者结合的理论进展。本文将此类结合称为张量神经网络（TNNs），并从网络压缩、信息融合和量子电路仿真三个主要方面介绍张量神经网络。此外，本综述还探讨了改进张量神经网络的方法，考察了用于实现张量神经网络的灵活工具箱，记录了张量神经网络的发展历程，并展望了潜在的研究方向。据我们所知，这是首篇全面连接神经网络、张量网络和量子电路之间关系的综述。我们在 \url{https://github.com/tnbar/awesome-tensorial-neural-networks} 提供了精心整理的张量神经网络资源列表。