I’m wondering if the bond dimensions in MPS is a good measurement of entanglement (and also some mechanisms of iTensor)
The reason I’m asking this, is because I did a TEBD calculation of my system of about 140 steps, and I can extrapolate roughly a \sim \exp(0.26t) scaling of bond dimension w.r.t time
If this is solely an artifact of the physics, it would indicate some O(t) linear growth of entanglement in time in my system.
However, for this calculation I did not restrict the max bond dimension, and was wondering if this could simply be a case of repeatedly applying the sets of (local and swap) gates, and somehow locally extending the state space, i.e. result of some numerical operation instead of physics.
Alternatively, what happens if cutoff and maxdim are both specified for apply()? I assume the maxdim takes priority.
(By looking at the source code for the apply() and contract() functions, it seems some eigen and/or SVD operations are performed each time, so I would tend to believe this is something related to the physics in the system, still it would be very helpful to be clarified on this issue)
I’m not sure I understand all the aspects of your question since it sounds rather open ended, or like a few different questions combined together. But basically yes it’s known from the tensor network literature that the bond dimension grows exponentially as a function of time when evolving a closed system. (At least in the generic case, like of a global quantum quench; it can grow more slowly in specific cases like a local quench on top of the ground state, say.)
So I would say that seeing exponential growth of the bond dimension is indeed because of the physics, but I would not call it an “artifact” as if was something to be removed. It is simply how generic closed quantum systems behave.
Also yes bond dimension is well understood to grow exponentially with entanglement, so linear growth in entanglement translates into exponential growth in bond dimension
Regarding your second question about how cutoff and maxdim work, it’s a good question. You can think of it as the cutoff being applied first, then if the resulting bond dimension is still greater than maxdim, that next maxdim is applied.
Might be an irrelevant question, and because the increase in entanglement is physical, I’m assuming different algorithm will eventually come to the same problem, but in this case, do you expect TDVP to at least perform better than TEBD? I’ve heard some arguments about TDVP having bigger ‘errors’ than TEBD, but I’ve not seen too much concrete evidence.
I ask because my system has some long-range interaction that decays as power law, and in this paper
They had a comment about “Unfortunately, the TEBD idea is not applicable to
long-range interactions”, but in your paper
and the iTensor package, LR interaction can still be simulated using swap gates. I’m wondering if they were just referring to the inefficiency of such set up? (although having O(L^2) number of swap gates, should most of them simply cancel with each other?)
As far as I know, TDVP let you keep your bond dimensions in some way, but since the two site algorithm performs SVD as well (which I assume happens for the TEBD contractions), I don’t see the different there.
The other thing I could think of is that for the Trotter decomposition you have to take small steps to minimize the errors, but TDVP doesn’t have such constraint. This is the main difference I can think of.
Sorry for the long question (and my lack of further understanding), I guess this is not really relevant to the package itself but as a more general question.
Overall it’s a complicated discussion about the pros and cons of different time evolution methods for tensor networks, but as a rule of thumb, currently TDVP is in general the best method for time evolving MPS, assuming that you get it to converge properly (i.e. assuming it does not get “stuck”, which can happen especially for 1-site TDVP if you don’t use a subspace expansion technique with it).
From my knowledge of these methods, here are some key points to answer your various questions:
one-site TDVP without a subspace expansion only uses a fixed bond dimension throughout, so there’s no concept in that case of growing bond dimension, but then also errors can grow because of the fixed bond dimension
one-site TDVP with a subspace expansion and two-site TDVP do automatically grow the bond dimension, and do so in a fairly optimal way, similar to TEBD
TDVP generally incurs lower bond dimensions throughout the calculation compared to TEBD with swap gates, because during the swap gate procedure, the sites can end up in a sub-optimal ordering and entanglement is highly sensitive to the site ordering. In TDVP the sites always stay in the same ordering.
TDVP uses highly accurate time evolution methods in the core step (Krylov, Runge-Kutta, etc.) which allow one to control time-step errors better and take larger time steps. So this can make it both more accurate and more efficient at the same time.
So overall TDVP can be faster, more accurate, and more able to deal with longer-ranged Hamiltonians than TEBD. The main advantages of TEBD is that it can be simple to code and understand, and doesn’t ever have the problem of “getting stuck” and silently giving wrong results like TDVP can. But using either 2-site TDVP, or a subspace expansion technique, or both together can often keep TDVP from getting stuck. As with any numerical method it’s important to check against exact results (free fermions, etc.) and on small systems to make sure a method is working reliably for your particular case.