WIP: implementation of DiagonalTensorMap

[Jutho]

I think we do want to have a DiagonalTensorMap type for the singular values and eigenvalues in the corresponding factorisations. Changing those tensor types arguably constitutes a breaking change, so we do not want to do this after releasing 1.0.

In this first PR, I will just implement the type (finished) and a minimal implementation with some tests, but not yet start to use it.

Switching svd and eig, eigh to use DiagonalTensorMap will constitute a second PR.

[lkdvos]

Looks very reasonable to me!
I would think that definitely here, we can probably get an efficient block iterator quite easily, which might alleviate the overhead of computing the block location, but for now, it might not even show up in profilers anyways.

[Jutho]

Looks very reasonable to me!
I would think that definitely here, we can probably get an efficient block iterator quite easily, which might alleviate the overhead of computing the block location, but for now, it might not even show up in profilers anyways.

Yes I agree the current blocks is not optimal, as it recomputes the offset for every block separately. One could write an iterator, or actually make that one return a SectorDict, as in many cases people might actually be interested in having all of them available at the same time, e.g. to make a plot of the entanglement spectrum.

[Jutho]

Ok, if all tests work, I think this is getting more or less to a final stage for now.

[lkdvos]

Before merging, as a feature request that will be useful for PEPSKit, powers of DiagonalTensorMap with real exponents would be nice to define here (along with the rrules). (https://github.com/QuantumKitHub/PEPSKit.jl/blob/90375ca9b34b8c3fc2c7e3550cbdfb033163ccff/src/utility/util.jl#L26)
For negative exponents, it would also be convenient to have a function that has some kind of cutoff, but I am not sure if such a function already exists for matrices even? pseudopower?

[Jutho]

Before merging, as a feature request that will be useful for PEPSKit, powers of DiagonalTensorMap with real exponents would be nice to define here (along with the rrules). (https://github.com/QuantumKitHub/PEPSKit.jl/blob/90375ca9b34b8c3fc2c7e3550cbdfb033163ccff/src/utility/util.jl#L26)
For negative exponents, it would also be convenient to have a function that has some kind of cutoff, but I am not sure if such a function already exists for matrices even? pseudopower?

Checking the matrix implementation, it is another one of those that is type unstable. For a real matrix, if p is integer, or A is positive definite, then the result is real, otherwise it becomes complex. What is furthermore funny, is this:

julia> A^0.57
2×2 Symmetric{ComplexF64, Matrix{ComplexF64}}:
  0.121409+0.155758im  -0.251072+0.107136im
 -0.251072+0.107136im   0.313729+0.0736924im

julia> Diagonal(eigvals(A))^0.57
ERROR: DomainError with -0.07888175758153752:
Exponentiation yielding a complex result requires a complex argument.
Replace x^y with (x+0im)^y, Complex(x)^y, or similar.
Stacktrace:
...

Because of all of this, I would very much like to postpone this to a separate PR.

[lkdvos]

I'm definitely okay with having this as a separate PR.

The results with the Diagonal you show are however exactly the behavior I would expect, and I would be completely okay with that throwing a DomainError. The cases where PEPSKit uses this, the diagonal tensor is the result of an SVD, so it shouldn't be a problem there.

[codecov]

Codecov Report

Attention: Patch coverage is 83.25123% with 34 lines in your changes missing coverage. Please review.

Project coverage is 81.39%. Comparing base (cf796ae) to head (0111bc9).
Report is 2 commits behind head on master.

Files with missing lines	Patch %	Lines
src/tensors/diagonal.jl	83.05%	30 Missing ⚠️
src/tensors/indexmanipulations.jl	66.66%	3 Missing ⚠️
src/tensors/braidingtensor.jl	0.00%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #174      +/-   ##
==========================================
+ Coverage   80.93%   81.39%   +0.45%     
==========================================
  Files          41       42       +1     
  Lines        5015     5218     +203     
==========================================
+ Hits         4059     4247     +188     
- Misses        956      971      +15     

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[Jutho]

Ok, thanks for the extra additions @lkdvos . I think this is now ready to be merged.