Fix Multiline (MultilineMPS/MultilineMPO) correctness and API inconsistencies

[borisdevos]

[Relevant edit deeper in conversation]

I found a cluster of bugs that all traced back to the same root cause, namely Multiline{T} was inconsistent about what it is. Sometimes it was treated as a 2D array indexed by (row, col), and sometimes as a 1D sequence of T-typed lines, this while different parts of the code picked different conventions without checking consistency. The real bugs were:

• MultilineMPO * MultilineMPS had never worked (straight up method error), and fixing that trivially still left both a shape bug (length counting rows*cols while map/collect only ever produced rows elements) and a silently wrong row-shift bug (fidelity 0 where it should be 1).
• expectation_value(::MultilineMPS, ::MultilineMPO, ...) had a fallback method that silently returned something meaningless (prod instead of sum, no row shift, discarded envs which was noted through a todo) for any MultilineMPO line type it wasn't explicitly guarded against. Related to this, there was half-support for Hamiltonian or finite MPO multilines even though they're meaningless. Also the environments weren't being correctly converted, so they were being recalculated every row.
• isfinite, changebonds(::MultilineMPO, ::SvdCut), and axes(m, 2) all threw outright.

A summary of the changes I made:

"Semantics":

• Multiline{T} now consistently treats length/eltype/iterate/m[i] as referring to the lines (matching the periodic rows), while size/axes/eachindex refer to the (nrows, ncols) lattice shape. These two views are intentionally different (length(m) != prod(size(m))). I also documented this in docstrings.
• Fixed multiplication: both were broken. The row-shifting is implemented and documented.
• Deleted the expectation_value fallback. Changed the infinite MPS-MPO expectation value to correctly account for environments.
• Fixed isfinite, changebonds(::MultilineMPO/::MultilineMPS, ::SvdCut), and axes(m, i).
• Some grassmann code needed a change related to the 1d vs 2d view thing above.
• Added the missing instance-level spacetype/sectortype/storagetype for multiline. I needed this for the custom show (see below).

Construction restrictions:

• MultilineMPO only accepts InfiniteMPO lines now, enforced at constructor level. This is deliberate, since Multiline is also used via converts in environments, quasiparticles etc.
• Removed the AbstractMatrix constructor that silently built finite-line MultilineMPOs, and the dead code in orthoview.jl that existed only to support finite MPS multilines.

Display:

• MultilineMPS/MultilineMPO custom show: added a summary which looks at the 2d shape, and a show that renders each row using that row's own existing show method. Here below an example of how it looks now:

Here the multiline MPS:

Details

2×2 MultilineMPS(ComplexF64, Vect[IsingAnyon]) with maximal dimension 10.0:
row 1:
2-site InfiniteMPS(ComplexF64, Vect[IsingAnyon]) with maximal dimension 10.0:
| ⋮
| (:σ => 7)
├─[2]─ (:σ => 1)
│ (:I => 5, :ψ => 5)
├─[1]─ (:σ => 1)
│ (:σ => 7)
| ⋮

  ⋮
row 2:
2-site InfiniteMPS(ComplexF64, Vect[IsingAnyon]) with maximal dimension 10.0:
| ⋮
| (:I => 5, :ψ => 5)
├─[2]─ (:σ => 1)
│ (:σ => 7)
├─[1]─ (:σ => 1)
│ (:I => 5, :ψ => 5)
| ⋮

And here a multiline MPO:

Details

2×2 MultilineMPO(ComplexF64, Vect[IsingAnyon]) with maximal dimension 1.4142135623730951:
row 1:
2-site InfiniteMPO(ComplexF64, Vect[IsingAnyon]) with maximal dimension 1.4142135623730951:
| ⋮
| (:σ => 1)
┼─[2]─ (:σ => 1)
│ (:σ => 1)
┼─[1]─ (:σ => 1)
│ (:σ => 1)
| ⋮

  ↓  (row 1 maps onto row 2)
row 2:
2-site InfiniteMPO(ComplexF64, Vect[IsingAnyon]) with maximal dimension 1.4142135623730951:
| ⋮
| (:σ => 1)
┼─[2]─ (:σ => 1)
│ (:σ => 1)
┼─[1]─ (:σ => 1)
│ (:σ => 1)
| ⋮

  ↓  (row 2 maps onto row 1)

Tests:

• New test/operators/multilinempo.jl: exact-fidelity tests for the row-shift convention in both MultilineMPO * MultilineMPS and MultilineMPO * MultilineMPO, using a permutation MPO whose fixed point is known exactly.
• test/algorithms/statmech.jl: multi-row expectation_value regression test. Also a test that the Hamiltonian loophole above still correctly throws.
• test/states/multilinemps.jl: added instance-level isfinite(ψ) (only the type-level form was covered before).

Documentation:

• docs/src/man/states.md: added the row-shift convention, what's currently supported (infinite lines only), and subtleties.
• docs/src/man/operators.md: new MultilineMPO section covering the same convention, and emphasising the MPO*MPO non-composability caveat.
• Updated docstrings on Multiline, MultilineMPO, MultilineMPS, and expectation_value.
• Added to the changelog, as the PR template wished of me 😄 .

Some things I didn't address because I'm not sure what to do with them:

• VectorInterface stuff is still broken for MultilineMPS, simply because InfiniteMPS itself doesn't implement zerovector/scale.
• norm(ψ) == sqrt(nrows) for a MultilineMPS of individually-normalized rows is unchanged. This goes through dot/inner sum over rows. I don't know if this is what you want.

Checklist

• Tests pass locally (julia --project=test test/runtests.jl, or the relevant subset)
• Documentation updated, if this PR changes public API (docstrings, docs/src/)
• Runic formatter is run
• Changelog entry added under [Unreleased] in docs/src/changelog.md, if this PR is user-facing (new feature, behavior change, bug fix, deprecation, or removal)

[codecov]

Codecov Report

❌ Patch coverage is 39.60396% with 61 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/utility/show.jl	0.00%	36 Missing ⚠️
src/states/orthoview.jl	0.00%	8 Missing ⚠️
src/utility/multiline.jl	50.00%	5 Missing ⚠️
src/algorithms/expval.jl	50.00%	4 Missing ⚠️
src/algorithms/changebonds/svdcut.jl	60.00%	2 Missing ⚠️
src/operators/multilinempo.jl	80.00%	2 Missing ⚠️
src/algorithms/grassmann.jl	75.00%	1 Missing ⚠️
src/algorithms/statmech/vomps.jl	75.00%	1 Missing ⚠️
src/environments/multiline_envs.jl	50.00%	1 Missing ⚠️
src/states/multilinemps.jl	75.00%	1 Missing ⚠️

Files with missing lines	Coverage Δ
src/MPSKit.jl	100.00% <ø> (ø)
src/algorithms/changebonds/randexpand.jl	63.63% <100.00%> (ø)
src/algorithms/groundstate/vumps.jl	98.70% <100.00%> (+0.01%)	⬆️
src/algorithms/statmech/idmrg.jl	98.51% <100.00%> (ø)
src/algorithms/grassmann.jl	79.03% <75.00%> (ø)
src/algorithms/statmech/vomps.jl	87.93% <75.00%> (ø)
src/environments/multiline_envs.jl	80.35% <50.00%> (-1.13%)	⬇️
src/states/multilinemps.jl	79.54% <75.00%> (-2.68%)	⬇️
src/algorithms/changebonds/svdcut.jl	89.18% <60.00%> (ø)
src/operators/multilinempo.jl	69.56% <80.00%> (+29.56%)	⬆️
... and 4 more

... and 1 file with indirect coverage changes

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[borisdevos]

After having discussed with @leburgel, I realised I didn't fully appreciate/understand what multiline actually represents, so recent changes rectify some mistakes I made, and tried to explain these niche things more cleanly in the docs/docstrings.

• I now allow finite lines at type and constructor level, but there's still no support for these at the algorithmic level. Hamiltonians are still correctly rejected.
• Multiplying multiline things never made sense, so I removed these methods and added the one that does make sense.
• Some orthoview changes I previously removed, plus fixing bounds checking on these views.
• I took the liberty of not calling the value that you would calculate with multiline an expectation value, but rather what it is, a dominant eigenvalue. So I introduced a new function dominant_eigenvalue, explained why it's separate from expectation_value and when these actually collide. The tricky thing here was that the VUMPS code worked for both MPOs and Hamiltonians, so there's a funky check there to see whether expectation values or dominant eigenvalues are being logged.
• Docs and docstrings clarifications.