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.
It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.
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.
MPS are ansatz wavefunctions that need to be optimized to describe the ground-state of a given Hamiltonian.
DMRG is one of the best method we have to optimize the MPS. Therefore you can think of MPS as a framework, and DMRG as an algorithm. Of course, this is not how things where developed historically, but that is the current reinterpretation.
I think mps is the inner structure of DMRG. And also mps is the reason why DMRG can succeed for its catching low entanglement of 1D systems. While considering systems in PBC conditon, VMPS can acheive much better results than DMRG. In some way, I think DMRG is nothing more than a special version of VMPS approach. DMRG's theory lies in mps.