community, thus giving birth to the time dependent DMRG , t-DMRG for short. Around the same time, other groups have developed similar approaches in which quantum information plays a predominant role as, for example, in DMRG implementations for periodic boundary conditions , and for studying mixed-state dynamics in one-dimensional quantum lattice systems,. Those last approaches actually provide a formalism that is more general than the original TEBD approach, as it also allows to deal with evolutions with matrix product operators; this enables the simulation of nontrivial non-infinitesimal evolutions as opposed to the TEBD case, and is a crucial ingredient to deal with higher-dimensional analogues of matrix product states. The decomposition of state Introducing the decomposition of State Consider a chain of N qubits, described by the function . The most natural way of describing would be using the local -dimensional basis : where M is the on-site dimension. The trick of TEBD is to re-write the coefficients : This form, known as a Matrix product state, simplifies the calculations greatly. To understand why, one can look at the Schmidt decomposition of a state, which uses singular value decomposition to express a state with limited entanglement more simply. The Schmidt decomposition Consider the state of a bipartite system . Every such state can be represented in an appropriately chosen basis as: where are formed with vectors that make an orthonormal basis in and, correspondingly, vectors , which form an orthonormal basis in , with the coefficients being real and positive, . This is called the Schmidt decomposition (SD) of a state. In general the summation goes up to . The Schmidt rank of a bipartite split is given by the number of non-zero Schmidt coefficients. If the Schmidt rank is one, the split is characterized by a product state. The vectors of the SD are determined up to a phase and the eigenvalues and the Schmidt rank are unique. For example, the two-qubit state: has the following SD: with On the other hand, the state: is a product state: Building the decomposition of state At this point we know enough to try to see how we explicitly build the decomposition (let's call it D). Consider the bipartite splitting . The SD has the coefficients and eigenvectors . By expanding the 's in the local basis, one can write: The process can be decomposed in three steps, iterated for each bond (and,