Hand-waving and Interpretive Dance

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file. Acknowledgements We thank everyone who stayed awake for the presentation of these lectures in January of 2016 at the Australian Institute of Nanoscience. We thank Stephen Bartlett, Andrew Doherty, Christopher Granade, Robin Harper, Marco Tomamichel, Dominic Williamson, and especially Doriane Drolet and David Tuckett, for their input. For suggesting that we give these lectures and editorial assistance, we give special thanks for the best-selling childrens’ author1 Chris Ferrie. We acknowledge support from the Australian Research Council via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013. Much of the material was reproduced from memory after one of the authors attended the Tensor Network Summer School at Universiteit Gent in 2015.