Add canonize function

[jofrevalles]

Summary

This PR resolves #22 by implementing the canonize function, which takes an input Chain and transforms it into the canonical form (also known as Vidal form), where we have a set of Γ and Λ tensors. For now, this function is implemented for Chain with Open boundaries.

Additionally, we added some tests to cover for the new function. Thanks to @starsfordummies for the useful comments!

Example

Let's show how this function works:

julia> qtn = rand(Chain, Open, State; n=5, p = 2, χ=20)
MPS (inputs=0, outputs=5)

julia> canonized = canonize(qtn)
MPS (inputs=0, outputs=5)

julia> length(tensors(canonized)) # 5 tensors + 4 singular values vectors
9

julia> isapprox(contract(transform(TensorNetwork(canonized), Tenet.HyperindConverter())),
                contract(TensorNetwork(qtn)))
true

julia> Γ₃ = select(canonized, :tensor, Site(3))
4×2×4 Tensor{Float64, 3, Array{Float64, 3}}: ...

julia> Λ₄ = pop!(TensorNetwork(canonized), select(canonized, :between, Site(3), Site(4)))
4-element Tensor{Float64, 1, Vector{Float64}}:
 0.8303349576858878
 0.4869563863594485
 0.22360345538771664
 0.15303212266448157

julia> sum((i -> i^2).(Λ₄))
1.000000000000001

julia> replace!(TensorNetwork(canonized), Γ₃ => contract(Γ₃, Λ₄; dims=()))
TensorNetwork (#tensors=8, #inds=9)

julia> @test Qrochet.isrightcanonical(canonized, Site(3))
Test Passed

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 75.00%. Comparing base (eb61105) to head (3c89308).

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #27      +/-   ##
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+ Coverage   72.94%   75.00%   +2.05%     
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  Files           8        8              
  Lines         414      436      +22     
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+ Hits          302      327      +25     
+ Misses        112      109       -3     

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[jofrevalles]

Question for @bsc-quantic/tensor-networks: should we do that the isleftcanonical, isrightcanonical functions already process the singular values?

I think that makes sense that we do not (so like it is currently in the code), and just take the tensor at that site and do the checking. This makes the tests for this function a little bit cumbersome, but do the work. In my opinion, we should add a function that does the contraction of the singular values, since I think this will be necessary for many algorithms.

[mofeing]

I think you no longer need to do the Tenet.HyperindConverter transformation to be able to contract. Would you mind updating your Tenet installation and trying it?

[jofrevalles]

I now tried using Tenet v0.5.2 and it did not work without the Tenet.HyperindConverter.

[mofeing]

First batch of requests. There is one thing that you don't test and I think it's very important: the norm of the TN. The L2-norm of each of the singular value vector on each bond should give the same value, and equal to contract(tn).

[jofrevalles]

@mofeing The code now will error due to the Tenet.HyperindConverter thingy. I have tested locally and everything works otherwise.

[jofrevalles]

Okay @mofeing, every test now has passed correctly. I would like to merge this.

[mofeing]

??