The mathematics and science behind Matter Modeling

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Q: Why does the Wolff algorithm slow down in a 4-body Ising model?

In the paper that introduced "Self-learning MC" (an ML-inspired MC technique, as I understand) the authors consider a many-body Ising model as an example to show the efficiency of their algorithm. The model looks like this: \begin{equation} H= -J \sum_{\langle i,j\rangle} S_i S_j - K \sum_{ijkl\i...

Has recently been discussed a bit in the hbar chatroom. I'll bring some of those chat messages here!

in The h Bar, 2 hours ago, by Nike Dattani

This question seems very interesting but seems to have been way to hard for anyone to answer so far. Perhaps it should have been asked at Physics.SE instead, does anyone here know anything about this topic?

in The h Bar, 54 mins ago, by fqq

It sure is interesting. Not knowing much about it I would guess Wolff's algorithm deals with two-point correlations better than with four-point ones. "it's not that different" is a bit of a stretch.
Do we know how well the algorithm performs in the plaquette-only ($J=0$) model?

in The h Bar, 34 mins ago, by Nike Dattani

@fqq I'm not sure about how it performs for the plaquette-only model, I'm curious to see what OP says about that! And that's a good suggestion about 2-point correlations being dealt with better than 4-point correlations, in fact you might have hit the nail perfectly on the head there?

in The h Bar, 28 mins ago, by fqq

The ML algorithm performs well, it's Wolff's algorithm (flipping clusters instead of spins) that gets slow. Still, clusters are built on links and not on plaquettes, maybe that's the point.
Actually, having a look at the paper, the ML algorithm seems to only learn two-point correlations (with different ranges) and still work well

in The h Bar, 22 mins ago, by Nike Dattani

@fqq You seem to have quite a grasp of what's going on in the Wolff's algorithm paper! It's been one of the longest standing unanswered questions on MMSE, so you'd probably get the necromancer badge if you do manage to answer it.

in The h Bar, 20 mins ago, by fqq

I'll definitely think about it, but at the moment I don't have a clear enough idea to answer it

in The h Bar, 18 mins ago, by Nike Dattani

@fqq We may have to wait for the OP to answer your comment first!

@Tyberius Thanks for removing the first message. It seems that a lot of the other initial messages from "feeds" about the room being frozen and unfrozen, could also be deleted to make things look less aesthetically unpleasing?

Hi @Jun_Gitef17!

fqq

5:20 PM

Hi all

Nike Dattani

Hi @fqq, I invited the OP (@Jun_Gitef17) of the Wolff algorithm question here, so that the question can be discussed in a room that has fewer distractions than the hbar!

Jun_Gitef17

Hi, thanks for the discussions. It's actually past 2 in the morning here in Japan and i need to go to be dsoon

Nike Dattani

bed* soon

Jun_Gitef17

It's been quite long since I asked the question so let me recap..

Nike Dattani

Yea I was wondering why you were up so late. First I thought it might have been the APS March Meeting, but that finished on the 19th!

You're now back in Japan? Did you get a new job now?

My understanding is that most algorithms are slower for Hamiltonians that have multi-spin interactions rather than just 2-body interactions (for example Helmut Katzgraber told me his software can treat multi-qubit interactions but it would slow things down), but I've never implemented any of these algorithms.. only worked on the k-local to 2-local transformation part.

Jun_Gitef17

5:31 PM

Just had a long discussion. I think I told you that I'm going to the US from next month!

About the question, I guess it's always possible to just say that "multi-spin interactions usually makes things slower". I accept that as a general trend. I wanted an intuitive explanation that differentiates 2-body from rest of the multi-spin interactions, if there is such a thing.
The situation seemed stranger for me when I realized that with or without the 4-body term, the algorithmic implementation nor the field-theoretic description of the Ising model doesn't change.

I actually had a try with the Wollf algorithm with the J=0, K=1 case.
It worked OK, but I haven't done a precise comparison with the two-body case.

Nike Dattani

5:59 PM

Oh yea I remember you telling me about the new job at the "Institute for non-stoquastic Hamiltonians" :)

"I wanted an intuitive explanation that differentiates 2-body from rest of the multi-spin interactions" -- maybe it's that 2 is smaller than all numbers larger than 2, hahaha!

@Jun_Gitef17 A detailed comparison would be interesting: one one hand it might still be slower than the 2-body case, if it's true that plaquettes always slow us down, but on the other hand it might be faster since it's only doing plaquettes and doesn't have to worry about the mixture of plaquettes and 2-body couplings. I don't think the plaquette-only case will be faster than the 2-body-only case.

Let's find out the result and then we can all publish a paper together if the result turns out to be surprising or interesting!

@fqq Were you expecting the plaquette only version to be faster?

fqq

6:39 PM

@NikeDattani I don't know. It's fast with 2-only, and "slow" with 2+4. If it's slow with 4-only we can probably think just about that. If it's fast with 4-only, the problem is somehow in the interplay between the two terms.

Nike Dattani

@fqq I doubt it would be fast for 4-only, and I think 4-only would be slower than 2-only but faster than 2+4, but let's wait until Jun wakes up!

This assumes that the number of terms remains the same. One 4-only term would be faster than 6000 2-only terms.

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The mathematics and science behind Ma…