[WIP] Add support for local expectation values of tensor product and MPO terms

[leburgel]

Built on top of #423.

Adds support for alternative operator forms for situations that are very badly suited to our current dense-tensor-operator approach, while still being compatible with AD to use them in variational optimizations. For now this is mostly a demonstration of how #423 allows to easily add support for the evaluation of different operator types by only overloading operator_contraction_expr. The concrete design will be revisited later.

[Yue-Zhengyuan]

I suppose this is exact contraction with CTMRG/BPEnv ?

[codecov]

Codecov Report

❌ Patch coverage is 95.77465% with 3 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/operators/localoperator.jl	95.65%	2 Missing ⚠️
...lgorithms/contractions/local_patch/network_expr.jl	93.33%	1 Missing ⚠️

Files with missing lines	Coverage Δ
.../algorithms/expectation_value/expectation_value.jl	91.30% <100.00%> (+0.82%)	⬆️
...algorithms/expectation_value/patch_contractions.jl	100.00% <100.00%> (ø)
...lgorithms/contractions/local_patch/network_expr.jl	95.91% <93.33%> (-0.47%)	⬇️
src/operators/localoperator.jl	79.73% <95.65%> (+6.34%)	⬆️

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[leburgel]

I suppose this is exact contraction with CTMRG/BPEnv ?

Yes, precisely that. The idea was to add support for alternative operator forms for situations that are very badly suited to our current dense-tensor-operator approach, while still being compatible with AD to use them in variational optimizations. In particular, the motivating cases are multi-site interactions that factor into a plain tensor product across sites (e.g. plaquette terms in gauge theories), and short-range two-site interactions for systems with a large physical dimension. Both of these are extremely expensive for our current approach, while simply storing the tensor product factors separately or storing the operator as a 2-site MPO of low bond dimension greatly reduces the cost.

This is still very much a draft, but I already filed the PR to demonstrate how #423 allows to easily add support for exact evaluation of different operator types. In particular, the MPOTerm here is a straight duplicate of the MPOObservable introduced in #415. This PR can just stay in draft status until the design that is merged, and then I'll adopt the resulting design here.

[Yue-Zhengyuan]

This will bypass the check in add_terms! and add two MPOs incorrectly. The issue can be detected by

Hₓ = LocalOperator(lattice, sites => gate_to_mpo(X ⊗ X))
Hᶻ = LocalOperator(lattice, sites => gate_to_mpo(Z ⊗ Z))
H = Hₓ + Hᶻ
@test dense_operator(only(values(H.terms))) ≈ X ⊗ X + Z ⊗ Z

where dense_operator is a function that converts the MPO back to a dense TensorMap.