# Difference between DMRG (density matrix renomalization group) and MPS (matrix product states)?

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?

• Modern DMRG codes are mostly written as vMPS, because vMPS is conceptually simpler. But of course which one is simpler is a subjective matter, to some extent. Nov 25 '15 at 16:29
• @MengCheng I think I understand these two approaches better now. MPS is the reason why DMRG can get such a good accuracy for its catching low entanglement of 1D systems. While VMPS is different with DMRG when simulating systems in PBC edge conditon. I think VMPS has more than DMRG. DMRG is a special version of MPS in some way.
– Ayu
Dec 2 '15 at 4:46

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.

• Welcome to Physics StackExchange! Note that we do have some Markdown you can use for emphasis, rather than capitalization. We also have Mathjax for equations. Mar 19 '16 at 16:54
• @EverettYou DMRG-X is not the only way to target MBL exited states. I have been using a self-made algorithm which uses Non-linear Conjugate Gradient to locally invert the state. The inverted state then has a ground state which can be specifically chosen by an initial shift of the Hamiltonian, and convergence is fast. If you want to use a CG like inversion you need sign-definite Hamiltonian which you can either achieve by using the square Hamiltonian or solving the normal equations. Hamiltonian is symmetric so, solving for the square Hamiltonian is enough and turns out to be more accurate. Mar 23 '16 at 12:25
• @AlexisMichailidis Sure, I did not say that DMRG-X is the only way to target MBL eigenstates. There have been a lot of methods developed last year, including MPS-base or RG-bases methods. Mar 23 '16 at 19:46

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.

• "DMRG is the optimal method to optimize the MPS": Why? Jan 3 '16 at 15:15
• @NorbertSchuch That's what I heard. But you are for sure the expert around here, and if my statement is wrong, I would happily modify my answer.
Jan 5 '16 at 9:42
• Sequential optimization of the tensors, as done in DMRG, is certainly a method which works very well in practice. However, I wouldn't claim it's optimal in general. (E.g., one can construct instances where it can get stuck.) I don't think we have a good idea what the optimal method would be (and, moreover, this might depend on the figure of merit). Jan 5 '16 at 9:55
• @NorbertSchuch: Ok, I wasn't sure if there was any proof that it is the optimal method. I'll change that by "one of the best method we have" then. Thanks.
Jan 5 '16 at 10:33
• It is not even clear the sequential optimization of MPS is finding the global optimum. (One can set up problems + initial conditions where such optimizations fail: arxiv.org/abs/quant-ph/0609051) Note that there is a provably converging algorithm which however (as of now, at least) is much less practical: arxiv.org/abs/1307.5143 Jan 5 '16 at 11:28

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.

• Historically MPS was discovered by inspecting the wavefunction obtained in DMRG, and when the full power of MPS was unravelled DMRG is reformulated with MPS. But even with MPS the PBC is hard to deal with -- although there is a very powerful method for infinite system, usually called iDMRG. Dec 2 '15 at 6:00
• "Historically MPS was discovered by inspecting the wavefunction obtained in DMRG" -- I (partly) disagree. MPS-like ansatzes were known before, such as the Finitely Correlated States of Fannes-Nachtergaele-Werner (but I'm pretty sure such ansatzes had been written down before). Dec 2 '15 at 8:04
• @MengCheng "But even with MPS the PBC is hard to deal with"--How about the VMPS by F.Verstraete(arXiv:cond-mat/0404706) and then improved by White(arXiv:0801.1947)? In these two papers they said that mps in PBC can get much better results than traditional DMRG even as good as traditional DMRG in OBC.
– Ayu
Dec 3 '15 at 8:49
• @NorbertSchuch I agree with you. MPS has been discovered before DMRG. But its power in simulating 1D quantum systems was found in DMRG.
– Ayu
Dec 3 '15 at 9:01
• @NorbertSchuch I agree, finitely correlated states were written down roughly around the same time as the original DMRG. But maybe it is fair to say that the importance of MPS in 1D was fully appreciated after the relation to DMRG was uncovered. Dec 3 '15 at 16:07