Belief propagation cache

[JoeyT1994]

This PR combines the partitionedITensorNetwork and message_tensors being passed around and used in belief propagation into one bpc::BeliefPropgationCache containing both. Relevant functionality is added for bpc such as getting message tensors, getting environments, updating message tensors (via belief propagation) and updating the tensors in the partitionedITensorNetwork.

Thus src/beliefpropgation/beliefpropagation.jl has essentially moved to src/caches/beliefpropagationcache.jl

test/test_beliefpropagation.jl has been refactored to reflect the new interface.

[codecov-commenter]

Codecov Report

Attention: Patch coverage is 86.24339% with 26 lines in your changes are missing coverage. Please review.

Project coverage is 73.61%. Comparing base (17132f2) to head (affa62c).
Report is 1 commits behind head on main.

Files	Patch %	Lines
src/caches/beliefpropagationcache.jl	83.16%	17 Missing ⚠️
src/gauging.jl	89.85%	7 Missing ⚠️
src/apply.jl	86.66%	2 Missing ⚠️

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Additional details and impacted files

@@            Coverage Diff             @@
##             main     #139      +/-   ##
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+ Coverage   73.59%   73.61%   +0.01%     
==========================================
  Files          70       71       +1     
  Lines        4117     4146      +29     
==========================================
+ Hits         3030     3052      +22     
- Misses       1087     1094       +7     

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[mtfishman]

@JoeyT1994 over all I think this is a big improvement, I like basing the BP code around the interface update(::BeliefPropagationCache) and I can see that this same kind of interface could extend to other cache types. For example, I could imagine extending this to update(::ContractionCache) corresponding to a cache for exact tensor network contraction, where the cache stores the tensor network, a contraction sequence, and a map from a subset of vertices to the result of contracting the subregion of the tensor network.

Then we can define contract(::ITensorNetwork; (cache!)::AbstractContractionCache) as a general interface for performing a contraction, where it uses the cache and updates the cache where appropriate (not saying we should do that in this PR, I think the scope of this PR is good as it is).

I've suggested a number of stylistic changes. One broad comment is that I think it would be good to change bpc to bp_cache, I kept forgetting what bpc stands for.

[JoeyT1994]

@mtfishman I am thinking about how best to improve the logic of this code based on your comments.

I think a helper constructor ITensorNetwork(psiv::VidalITensorNetwork) which absorbs all bond tensors onto the sites symmetrically (by taking square roots) and returns to the arbitrary gauge is going to be helpful.

Then you could have the constructor VidalITensorNetwork(psi::ITensorNetwork, bp_cache::BeliefPropagationCache) which constructs a VidalITensorNetwork via the algorithm described in our paper.

Then you could have update(psiv:VidalTensorNetwork; (cache!) = nothing, cache_update_kwargs) which first converts to the symmetric gauge as you suggested in your comments. Then does whatever level of BP that you desire, and then calls the constructor above. That way you never simultaneously have bond_tensors and a bp_cache

What do you think? I know that VidalITensorNetwork(psi::ITensorNetwork, bp_cache::BeliefPropagationCache) doesn't do any cache updating and just takes it at face value but I personally think that is okay and a logical function to have?

[mtfishman]

Thanks, definitely a nice improvement. Looking forward to seeing this in action in the alternating_update code!