Tensor networks capture natural quantum states of strongly correlated quantum systems well. In this talk, in line with the mindset of the inaugural meeting of the International Quantum Tensor Network, we will explore three comparably new lines of thought on the use of tensor networks. In the first part, we will see how random network networks allow for analytical and rigorous computations in settings that are otherwise hard to come by. This applies to the study of notions of generic quantum states of phases of matter [1], of critical states emerging in the context of holography [8] or of complexity [3]. Randomness can also be used to literally estimate entanglement in quantum many-body systems [4]. In a second part, we will have a look at new endeavours to make use of tensor networks to capture properties of real quantum materials in the laboratory, including ones aimed at observing many-body localization in quantum materials [5,8], or to augment tensor networks with mode transformations [6]. In the last part, we will see that tensor networks constitute powerful tools also in the context of learning tasks: This applies to scalable Hamiltonian learning from data [8] or the learning of classical dynamical laws, where tensor networks are made use of not in order to capture local quantum degrees of freedom, but in fact function dictionaries [7,8]. The meta-message of this overview talk will be to make the point that tensor networks remain powerful and inspiring tools to capture common structures in quantum and classical many-body theory.
[1] PRX Quantum 2, 040308 (2021).
[2] Quantum 6, 643 (2022).
[3] Nature Physics 18, 528 (2022).
[4] PRX Quantum 3, 030312 (2022).
[5] Ann. Phys. 421, 168292 (2020).
[6] Phys. Rev. B 104, 075137 (2021).
[7] arXiv:2002.12388, NeurIPS (2021).
[8] In preparation (2022).