Tensor network has been a promising numerical tool for computational problems across science and AI. For their emerging and fast development especially in the intersection between AI and science, this paper tries to present a compact review, regarding both their applications and its own recent technical development including open-source tools. Specifically, we make the observations that tensor network plays a functional role in matrix compression and representation, information fusion, as well as quantum-inspired algorithms, which can be generally regarded as Science4AI in our survey. On the other hand, there is an emerging line of research in tensor network in AI4Science especially like learning quantum many-body physics by using e.g. neural network quantum state. Importantly, we unify tensorization methodologies across classical and modern architectures, and particularly show how tensorization bridges low-order parameter spaces to high-dimensional representations without exponential parameter growth, and further point out their potential use in scientific computing. We conclude the paper with outlook for future trends.

We rethink the basic operations, i.e., inner product and matrix multiplication used in neural networks. A quantum-inspired alternative is proposed, utilizing the power of high-dimensional Hilbert space by devising a high-order form of tensor product. We re-parameterize the original (low-order) vectors/matrices into an expressive high-order form, without incurring extra model parameters, and the extra computational overhead is negligible (e.g., about 2%). As an in-place transparent atomic operation, we show its use in the key components in Transformers: token embeddings, attentions (query, key, value) and the MLP. Due to its inherent compatibility to vanilla multiplicative operations, we propose C2Q-SFT, i.e., classic-to-quantum (C2Q) protocol for supervised fine-tuning (SFT): it continues to train a given model by transparently replacing the standard operations with ours. As shown by our experiments, it shows advantages for both training from scratch and fine-tuning on downstream tasks across scales of LLMs. C2Q-SFT consistently outperforms standard SFT, with relative improvements on MMLU (+0.56%) and GSM8k (+0.61%). It sheds light on the innovation of operations in networks, orthogonal to the efforts on new architecture, position encoding, and training algorithms, etc. See project page at: https://github.com/Thinklab-SJTU/LLM/QI-LLM.