Add double-layer-pepo reduced densitymatrix

[lkdvos]

Follow-up for #250 and #184:

I want to write reduced density matrices for densitymatrix * densitymatrix' sandwiches, assuming that we'll obtain the CTMRG environments via fusing the InfinitePEPO into a InfinitePEPS construction.

• Write double-layer reduced densitymatrix contraction
• Write InfinitePEPO to fused InfinitePEPS converter
• Add tests

@Yue-Zhengyuan if you have test cases laying around that would be wonderful

[codecov]

Codecov Report

❌ Patch coverage is 85.52632% with 11 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/algorithms/contractions/localoperator.jl	94.00%	3 Missing ⚠️
src/networks/tensors.jl	50.00%	3 Missing ⚠️
src/operators/infinitepepo.jl	57.14%	3 Missing ⚠️
src/operators/localoperator.jl	85.71%	1 Missing ⚠️
src/states/infinitepeps.jl	50.00%	1 Missing ⚠️

Files with missing lines	Coverage Δ
src/algorithms/toolbox.jl	98.30% <100.00%> (+0.01%)	⬆️
src/operators/localoperator.jl	89.09% <85.71%> (+2.68%)	⬆️
src/states/infinitepeps.jl	64.65% <50.00%> (-0.26%)	⬇️
src/algorithms/contractions/localoperator.jl	78.57% <94.00%> (+4.10%)	⬆️
src/networks/tensors.jl	73.33% <50.00%> (-1.67%)	⬇️
src/operators/infinitepepo.jl	61.85% <57.14%> (+16.30%)	⬆️

... and 1 file with indirect coverage changes

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[Yue-Zhengyuan]

We can test the functions using test/examples/tf_ising_finiteT.jl in #250; just replace the current "manual" measurements with expectation_value. And energy measurement can also be added. (Though fermions haven't been covered)

The densitymatrix * densitymatrix' case corresponds to the purified approach to get exp(-βH). But if we don't care about it being Hermitian, it is also useful to have the reduced density matrix constructed from a single layer of PEPO (densitymatrix itself), with a function signature like

reduced_densitymatrix(inds, ρ::InfinitePEPO, env::CTMRGEnv)

where env is for the InfinitePartitionFunction obtained from ρ by tracing out the physical legs. The cost for the latter approach is smaller because the PEPO bond dimension is D instead of D^2.

[lkdvos]

Yeah, I would definitely like to add both but this one was easier since it was more similar to the peps case. I’ll try generalize in a follow up PR if that’s ok?

[Yue-Zhengyuan]

OK. For the tests, besides the physical example of Ising model, we can just generate a random PEPO and CTMRGEnv (no need to converge the env), and check the measurement results are the same as what we would obtain if we fuse the virtual legs and the physical legs respectively of the PEPO.

[lkdvos]

That's a great point about the tests, thanks a lot, this really sped up the dev time for me.
I think this is now working and ready, as discussed I'll add the single-layer algorithm in a separate PR, to avoid having a huge PR here.

[Yue-Zhengyuan]

Do you mind if I make the tests a bit more strict (and some other minor changes)?

[lkdvos]

Feel free to make the changes, and I agree with your comments

[Yue-Zhengyuan]

OK to merge for me once the tests pass.