Tensor Networks, Matrix Product States and Density Matrix Renormalization Group

In this note, we introduce the notions of tensor networks and matrix product states (MPS). These objects are particularly useful in describing quantum states of low entanglement. We then discuss how to efficiently compute the ground states of the Hamiltonians of 1D quantum systems (using classical computers). The density matrix renormalization group (DMRG), due to White (1992, 1993), is arguably the most successful heuristic for this problem. We describe it in the language of tensor networks and MPS.