Defining a sampler with a specific transition rule

[waleed-sh]

Hi! My questions is regarding defining a sampler with a custom transition rule. I saw the code for the TwoLocal rule here, but I am still a little lost...

The idea is the following:
I am trying to define a local sampler which somehow only generates states which have a certain relation between different basis elements. For example, say I have a 5 site chain, so basis elements are [a, b, c, d, e]. I want the sampler to somehow only generate states in which, for example, c = -a, and/or b = - d - e.

In essence, I have a system with a very large Hilbert space, but I know what type of states I should be looking for so I was wondering if I can already narrow the space to be explored at the sampling level. Any help or guidance would be appreciated!

[PhilipVinc]

Can you formalise the rule you'd want mathematically? What is your local basis? A spin-1?
It's hard to parse what you mean exactly.

[PhilipVinc]

Are you able of writing a function fthat given a single basis state and the three indices I,j,k gives you all possible states that satisfy that condition?

[waleed-sh]

Yes I think this is something possible to do so it shouldn't be a problem.

[waleed-sh]

I was thinking about this again and maybe this issue could be solved by defining a constrained Hilbert space? So from the start only allow for states which obey a certain criteria? But from my understanding this can cause issues with samplers which do not respect this criteria so I am not sure if then I will have to either way construct a sampler which does the same thing. Or am I wrong?

[PhilipVinc]

What you want is equivalent to what HamiltonianRule does.

If you look at its source, it queries the operator for its connected entries (this could be replaced by your function computing the connected entries given one), then it picks one, then it checks the connected components of the proposed site to compute the Hastings log correction, as your proposal might not be simmetrical.

You'd want to replace calls to get_conn_flattened to calls to your own function that lists the connected components given a certain configuration.

--

I was thinking about this again and maybe this issue could be solved by defining a constrained Hilbert space

Unlikely. Hilbert spaces are used to generate random configurations (like when you initialise your Markov chains) or for full summation things. Samplers might drop out of the constraint if, indeed, they do not respect the constraint.
The solution is, as above, to write a transition rule that respects it.

[waleed-sh]

Thank you for your reply! Just a quick possibly stupid question, what would the return type and the arguments of the get_conn_flattened? Because it uses the get_conn_flattened_closure() from the operator, which eventually leads to the complicated get_conn_flattened_kernel() (if I understand correctly). I'm just trying to see what datatype should the function expect and what is the simplest thing it should return for it to function.

Thank you for your reply though! This has been tremendously helpful. I think I will go with both a constrained Hilbert space and a sampler respecting this constraint as I also face some issues due to the dimensionality of the space.

[PhilipVinc]

Don't look at that stuff, it's overly complicated to work around limitations of numba while maintaining high performance.

Look at this version

netket/netket/sampler/rules/hamiltonian.py

Line 89 in a1286d7

def _transition(v, rand_vec):

it's equivalent. Just a tad bit slower, but much easier to understand.

(t's taken from this PR #1747 )

[waleed-sh]

Great, thanks again! I will give it a try I think it should be straightforward enough.