NTU: approximate contraction of a finite part of iPEPS

[Yue-Zhengyuan]

In neighborhood tensor update (NTU), the environment of a bond can be a finite patch around the bond. For example, if we contain two more layers around a bond, the environment is (the bra is contracted with ket)

    (-2 -2)═(-2 -1)═(-2 +0)══(-2 +1)═(-2 +2)═(-2 +3)
       ║       ║       ║        ║        ║        ║
    (-1 -2)═(-1 -1)═(-1 +0)══(-1 +1)═(-1 +2)═(-1 +3)
       ║       ║       ║        ║        ║        ║
    (+0 -2)═(+0 -1)════Q0══   ══Q1═══(+0 +2)═(+0 +3)
       ║       ║       ║        ║        ║        ║
    (+1 -2)═(+1 -1)═(+1 +0)══(+1 +1)═(+1 +2)═(+1 +3)
       ║       ║       ║        ║        ║        ║
    (+2 -2)═(+2 -1)═(+2 +0)══(+2 +1)═(+2 +2)═(+2 +3)

This is called EnvApproximate of type 65 in YASTN documentation. May I ask @leburgel if MPSKit already have boundary MPS method to contract such a finite network? Alternatively, @pbrehmer how can I use CTMRG to contract it? (In my mind CTMRG is for infinite size, so it's not quite clear to me how to apply it to a finite system).

[leburgel]

Using MPSKit, you could contract the patch from the top to towards the open bond by using the approximate algorithm to sequentially approximately contract a finite MPO slice of the patch into the top boundary MPS and return another MPS, until you've contracted everything above the open bond. You could then do the same starting from the bottom (where you would have to be a bit careful with how you pass the MPS and MPO slices to the algorithm, since the algorithm always assumes an MPO to be contracted into an MPS from below, where here you contract it from above). After this you have just the MPO-like slice with the open bond, and a top and bottom finite boundary MPS, at which point you can contract everything manually.

So this is not really a plug-and-play built-in routine, but it can definitely be done using MPSKit.approximate.

[pbrehmer]

... how can I use CTMRG to contract it? (In my mind CTMRG is for infinite size, so it's not quite clear to me how to apply it to a finite system)

From what I've head, in principle one can also use CTMRG for contracting finite environments with reasonable accuracy but I have no experience with this. I would be skeptical if that would be a viable approach for small-ish finite environments like the one you're trying to contract. In any case, the current CTMRG implementation only works for infinite lattices. The boundary MPS technique Lander suggested seems to be the way to go, I think.

[Yue-Zhengyuan]

Indeed in YASTN they use the boundary MPS approach. I'll try to implement this feature in near future as an option to NTU (#144).

[Yue-Zhengyuan]

@leburgel Looking at the method signature of approximate:

approximate(ψ₀, (O, ψ), algorithm, [environments]; kwargs...) -> (ψ, environments)

I wonder what's a good way to obtain the initial guess ψ₀? Do you just use local SVD truncation on each bond? Actually I have one method based on some sort of "projectors" (it's exact for truncation of one bond on an OBC-MPS), but maybe you have a better approach.

[Yue-Zhengyuan]

In addition, do we already have a function to construct an InfinitePartitionFunction from $\langle \psi | \psi \rangle$ so I don't need to write again? To use the MPSKit functions it is easier to first merge the virtual legs of the bra network and the ket network.

[lkdvos]

I think I added a mpotensor function recently in PEPSKit that takes a sandwich and spits out the correct MPOTensor, which you can then use in MPSKit. In principle the goal would be to be able to work with the double-layer objects directly as well, but there are definitely some parts missing there so it can be a good short-term solution to map to a single-layer network first.

[Yue-Zhengyuan]

Suppose that for convenience, I merge the virtual indices in such a way that the two-layer cluster is converted to a single-layer network with the following choice of arrows:

    C ← E ← E --←-- E ← E ← C
    ↓   ↓   ↓       ↓   ↓   ↓
    E ← T ← T --←-- T ← T ← E
    ↓   ↓   ↓       ↓   ↓   ↓
    E ← T ← X ←   ← Y ← T ← E
    ↓   ↓   ↓       ↓   ↓   ↓
    E ← T ← T --←-- T ← T ← E
    ↓   ↓   ↓       ↓   ↓   ↓
    C ← E ← E ← E ← E ← E ← C

and I compress the cluster horizontally starting from the left and right boundary. The right boundary is a very standard ket MPS, and the column transfer MPO acts on its left. However, the left boundary is a bra MPS, and the column transfer MPO acts on its right.

It seems that adjoint is not defined for a FiniteMPS in MPSKit (although it is for FiniteMPO). @leburgel @lkdvos Can MPSKit actually handle a bra MPS? If not, can you suggest a workaround?

[lkdvos]

@leburgel do you happen to have an example laying around of doing the required mapping to go from ket to bra?

[leburgel]

I've definitely done this a few times, let me see if I can dig up an example when I find some time.