Simulations with matrix product states

Matrix product states (MPS) are considered to be the most efficient parametrization of quantum states that emerge as ground states or low-lying excitations of short-ranged Hamiltonians in one spatial dimension. The most powerful simulation algorithms for strongly correlated quantum systems in one dimension, the family of density-matrix renormalization-group (DMRG) algorithms, can be understood as acting in this very particular state class and find a particularly transparent formulation if expressed in terms of matrix product states. In this set of lectures, I will introduce matrix product states, discuss their properties, the typical quantum mechanical operations expressed in their language, matrix product operators, and present both static and dynamic simulation algorithms. I will also make a connection to elements of entanglement theory.