Tensor Networks

Tensor networks study how to represent large arrays, probability distributions, quantum states, and neural-network parameters as networks of smaller tensors linked by bond indices. Common forms such as matrix product states, tree tensor networks, and matrix product operators support compression, inference, scientific computing, and quantum-circuit simulation by exploiting low-rank structure. Researchers develop rank-selection and contraction algorithms alongside learning methods, including Riemannian optimization, tensorized neural networks, and hardware-aware implementations. A central tension is that compact representations can reduce memory and computation, but their accuracy and practical speed depend on the data’s structure, the chosen network geometry, and the costs of training and contraction.