Introduction to Tensor Network Methods

Timo Felser received his doctoral degree in Physics with distinction from the University of Padua and the University of Saarland, working on the development of tensor networks for high-dimensional quantum many-body systems. His research interests focus on tensor network development and applications in various fields, ranging from low-energy to high-energy physics up to medical physics and machine learning. He published his work in journals with high impact factor, such as Physical Review X, Physical Review Letters, Nature Comm., npj Quantum Information. Further, he developed tensor network computations to perform machine learning tasks in the field of AI and is now leading the research transfer project Tensor Solutions at Ulm University to spin-off the tensor network machine learning technology into a start-up which aims to address data problems in industry.Simone Montangero is Full Professor at the Department of Physics and Astronomy of University of Padova . He has been a Heisenberg Fellow of the German Science Foundation, and a Humboldt Fellow. He is Honorary Professor at Ulm University and an IQOQI visiting fellow of the Institute for Quantum Optics and Quantum Information of the Austrian Academy of Science. He is a member of the Quantum Coordination Board of the EU-Quantum Flagship and of the Scientific Council of the National Metrology Institute of Italy - INRiM. He co-coordinates the quantum activities of the Italian National Center for HPC, Big Data and Quantum Computing - Foundation ICSC. Prof. Montangero has pioneered the development of tensor network methods for quantum technologies and lattice gauge theories, and the application of optimal control theory to many-body quantum systems. He has published more than 170 research articles at the interface of quantum science, many-body physics, and condensed matter in international journals and books, including Science, PNAS, Physical Review Letters, Physical Review X, Nature Communication. He has been involved in many national and international projects on quantum science: as coordinator of the EU-QuantERA QTFLAG and T-NISQ and as principal investigator of the EU projects RYSQ, SIQS, DIADEMS, PASQUANS, PASQUANS2, EURYQA, QEC4QEA.

• Introduction.- Linear Algebra.- Numerical Calculus.- Numerical Renormalization Group Methods.- Tensor Network Methods.- Symmetric Tensor Networks.- Matrix Product States.- Tree Tensor Networks.- Augmented Tree Tensor Network.- Quantum Phase Transitions.- Hamiltonian Lattice Gauge Theories.- Out-of-equilibrium Processes.- Tensor Networks for Machine Learning.