Matrix Product State Algorithms: DMRG, TEBD and Relatives

Abstract

The density-matrix renormalization group (DMRG) has established itself as the leading algorithm for the simulation of one-dimensional strongly correlated quantum systems and has been extended in various directions, from the calculation of ground states to e.g. the real- and imaginary-time evolution of quantum states, or to algorithms attempting to simulate two-dimensional quantum systems. While the original formulation of the DMRG algorithms was framed in well-known concepts of statistical mechanics (density matrices, decimation and others), it turned out that it is in many circumstances much more powerful to view this class of algorithms as natural manipulations of very special quantum mechanical state classes, the so-called matrix product states. In this chapter, both time-independent and time-dependent algorithms are presented in this framework, based on an introduction to matrix product states. I will also address the connection between matrix product states and quantum entanglement, which defines both the potential and the limitations of this class of algorithms.