Introduction to Tensor Network Methods

This second edition of the textbook Introduction to Tensor Network Methods contains more advanced and technical parts as new topics related to tensor network algorithms that have been developed in the last few years. The reader finds new chapters dedicated to tree tensor networks for high-dimensional systems as applications to lattice gauge theory. The implementation of tensor networks for machine learning is also presented in detail.

This textbook gives an in-depth overview on the numerical simulation technique of tensor networks (TNs) with hands-on technical descriptions, work exercises and computation results. TNs have originally been developed for solving the quantum many-body problem and simulating quantum systems on a classical computer. However, as a mathematical tool, TNs have emerged as powerful theoretical and numerical versatile tools to attack more generally hard mathematical problems. In particular, their range application has expanded to combinatorial optimization and even as an alternative tool for machine learning in the field of artificial intelligence. This textbook introduces the reader to the field, describing the main principles and core mathematical concepts in the light of its application in quantum physics and, along the way, touches on the application of TNs to problems from various fields, ranging from low-energy to high-energy physics up to medical physics and machine learning.

It is designed for graduate courses in computational physics, where a student learns how to write a tensor network program and can begin to explore the physics of many-body quantum systems.

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.