# [Physics] Difference between DMRG (density matrix renomalization group) and MPS (matrix product states)

algorithmscondensed-matterquantum-informationrenormalizationtensor-network

I am learning DMRG recently. I noticed there are many papers both in the DMRG approach and MPS (such as variational matrix product state (VMPS) by F.Verstraete and J.I.Cirac) approach.
In my eyes, there is no deep difference between these two approaches. One question that can be simulated by MPS also can be done by DMRG. So, In practical computation in 1D systems, I believe DMRG is preferred for its simplicity.
Does mps approach have a typical advantage against DMRG in practical simulation?

Matrix product state (MPS) is a way to write down many-body quantum states. It's a natural representation for infinite 1D states that bipartite entanglement entropy obeys area-law ($S = constant$). This doesn't mean that it can't represent finite systems which are not 1D and $S = F(L)$, where L is some dimension of the system. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.
Density matrix renormalization group (DMRG) is an efficient method to find the optimal MPS representation of the many-body state. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. Recently there has been a lot of research progress along this direction, using DMRG method to target highly-excited (finite-energy-density) states in many-body localization (MBL) systems, and the method is now called DMRG-X (with "X" stands for excited states). It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.