Enhance contract to support multiple input Tensors

[jofrevalles]

This PR enhances the contract function for multiple tensors, which resolves issue #16 in EinExprs. The function utilizes reduce to perform pairwise Tensor contractions.
To ensure the robustness of these changes we added a testset that covers the new contract functionality, checking its behavior with various tensors in inputs.

[codecov]

Codecov Report

Merging #31 (c6c1312) into master (5c2dae2) will increase coverage by 0.07%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master      #31      +/-   ##
==========================================
+ Coverage   82.05%   82.12%   +0.07%     
==========================================
  Files           6        6              
  Lines         234      235       +1     
==========================================
+ Hits          192      193       +1     
  Misses         42       42              

Impacted Files	Coverage Δ
src/Numerics.jl	87.30% <100.00%> (+0.20%)	⬆️

[mofeing]

I prefer not to merge this PR. Reasons is that if you need to contract several Tensors while in a EinExpr, it's better to create a new EinExpr level.

Also, what if you have 3 tensors with 1 index? This implementation wouldn't work because the index would disappear after the first contraction.

In my opinion, it is semantically ambiguous what you want to achieve with contract(tensors...).

[jofrevalles]

The problem is that EinExprs does not return only pairwise Tensors to be contracted, and that leads to error. You can check the issue I mentioned above. I think that this extension of the contract function simply fixed those problems.

[mofeing]

The problem is that EinExprs does not return only pairwise Tensors to be contracted, and that leads to error. You can check the issue I mentioned above. I think that this extension of the contract function simply fixed those problems.

Yep, but a optimization pass can be written such that

EinExpr(a,b,c,out=labels)

gets transformed to

EinExpr(EinExpr(a,b,out=labels_ab),c,out=labels)

[mofeing]

Yep, but a optimization pass can be written such that

EinExpr(a,b,c,out=labels)

gets transformed to

EinExpr(EinExpr(a,b,out=labels_ab),c,out=labels)

@jofrevalles Actually, we don't need to write any other optimization pass: this should be currently done by just calling einexpr(EinExpr([a,b,c], out=labels), optimizer=Exhaustive()).

I mean, just calling einexpr on the EinExpr that you want to "optimize".

[jofrevalles]

@jofrevalles Actually, we don't need to write any other optimization pass: this should be currently done by just calling einexpr(EinExpr([a,b,c], out=labels), optimizer=Exhaustive()).

I mean, just calling einexpr on the EinExpr that you want to "optimize".

It should, but this doesn't work for unconnected tensors, since einexpr does not return them two-by-two:

julia> tensor_vector = [
                  Tensor(rand(2, 2), (:i, :j)),
                  Tensor(rand(2, 2), (:k, :l)),
                  Tensor(rand(2, 2), (:m, :n))]
3-element Vector{Tensor{Float64, 2, Matrix{Float64}}}:
...

julia> einexpr(Exhaustive, EinExpr(tensor_vector)) |> contract
ERROR: MethodError: no method matching contract(::Tensor{Float64, 2, Matrix{Float64}}, ::Tensor{Float64, 2, Matrix{Float64}}, ::Tensor{Float64, 2, Matrix{Float64}}; dims::Vector{Symbol})

Closest candidates are:
  contract(::Tensor, ::Tensor; dims)
   @ Tensors ~/.julia/packages/Tensors/O9RAx/src/Numerics.jl:36
  contract(::Tensor; dims)
   @ Tensors ~/.julia/packages/Tensors/O9RAx/src/Numerics.jl:49
  contract(::Tensor, ::TensorNetwork; kwargs...)
   @ Tenet ~/git/Tenet.jl/src/TensorNetwork.jl:480
  ...

That's why I think that in this case it would be useful to extend contract.

[mofeing]

Okay, I'm gonna approve this but keep in mind that this is subject to changes in the future.