The Many-Body Tensor Representation

Matthias Rupp

Guillermo Valverde, Igor García-Olaizola, Giannicola Scarpa +1 more

Tensor networks were developed in the context of many-body physics as compressed representations of multiparticle quantum states. These representations mitigate the exponential complexity of many-body systems by capturing only the most relevant dependencies. Due to the formal similarity between quantum entanglement and statistical correlations, tensor networks have recently been integrated in machine learning, operating both as alternative learning architectures and as decompositions of components of neural networks. The expectation is that the theoretical understanding of tensor networks developed within quantum many-body physics leads to novel methods that offer advantages in terms of computational efficiency, explainability, or privacy. Here we review the use of tensor networks in the context of machine learning, providing a critical assessment of the state of the art, the potential advantages, and the challenges that must be overcome.

Xiongjie Yu, David Pekker, Bryan K. Clark

Zohreh Gholami, Zahra Noorinejad, Mohsen Amini +1 more

Abstract A novel method has been devised to compute the Local Integrals ofMotion~(LIOMs) for a one-dimensional many-body localized system. Inthis approach, a class of optimal unitary transformations is deducedin a tensor-network formalism to diagonalize the Hamiltonian of thespecified system. To construct the tensor network, we utilize theeigenstates of the subsystems' Hamiltonian to attain the desiredunitary transformations. Subsequently, we optimize the eigenstatesand acquire appropriate unitary localized operators that willrepresent the LIOMs tensor network. The efficiency of the method wasassessed and found to be both fast and almost accurate. In frameworkof the introduced tensor-network representation, we examine how theentanglement spreads along the considered many-body localized systemand evaluate the outcomes of the approximations employed in thisapproach. The important and interesting result is that in theproposed tensor network approximation, if the length of the blocksis greater than the length of localization, then the entropy growthwill be linear in terms of the logarithmic time. Also, it has beendemonstrated that, the entanglement can be calculated by onlyconsidering two blocks next to each other, if the Hamiltonian hasbeen diagonalized using the unitary transformation made by theprovided tensor-network representation.

Tao Xiang, J. Chen, Xiangzhou Kong

Ingvar Lindgren