This lecture series is part of QIC 890/891 held at the University of Waterloo, Spring 2011 and organized by Michele Mosca. We will present our variant of the graphical notation of Penrose, tailored to describe networks of connected tensors representing quantum states, operators and processes, and as a uniting tool for the notation and language of quantum theory, information science and condensed matter physics.

Topics covered: Tensor networks applied to representations of algebras and groups, symmetries, symmetric states, polynomial or algebraic invariants. Hamiltonian models of quantum computation, entanglement spectrum, stabiliser and match gate tensors. Penrose graphical notation and Penrose wire bending map-state duality via string diagrams. Factorisation of quantum states into tensor networks. Matrix Product States (MPS), Projected Entangled Pair States (PEPS), Multiscale Entanglement Renormalisation Ansatz (MERA), Categorical Tensor Network States (CTNS), Boolean Tensor Network States (BTNS).

Lecture notes and links to the videos are found on the course homepage: http://qubit.org/courses/46-iqc2011.html