…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…o enable_new_opmath

…ian` (#5216)

Branching: #5269 >
#5216 >
#5322 >
#5335

The basic idea is to have a clone of `qml.Hamiltonian` with a better
name, `LinearCombination` - that inherits from `CompositeOp` and plays
nice with new opmath. The motivation is that sometimes it is still handy
to have an operator class for which you _know_ there is no funny nesting
and that has `coeffs` and `ops` separately on demand without processing.

ToDo

- [x] basic init
- [x] `tests/ops/op_math/test_linear_combination.py` pass
- [x] update map_wires
- [x] all tests pass
- [x] add pauli_rep
- [x] add operands attribute and make iterable (?)
- [x] update string repr
- [x] upgrade to internally use new opmath
- [x] update tests with default new opmath
- [x] dunder math methods support with new opmath
- [x] simplify and speed-up `simplify()`, also make sure to not act
in-place
- [x] Could we make the matrix generation workflow more general like
Sum?
- [x] Also, deferring to the pauli rep method for matrix generation
leads to substantial performance improvements.
- [x] toggle `__use_new_opmath`
- [x] tests_linear_combination pass
- [x] Integration tests
- [x] get all linear combination tests to pass locally
- [x] utilize super().sparse_matrix
- [x] use grouping functionality from Sum that Mudit is adding in
#5179
- [x] update matmul to handle other LinearCombination instances
- [x] remove top level import
- [x] remove unnecessary copies
- [x] add diagonalizing gates
- [x] bugfix trivial case of simplify empty LinearCombination
- [x] add trivial case for is_hermitian
- [ ] torch differentiation with simplify=True in constructor
- [x] changelog

Optional features (likely to be included in a follow-up)
- [x] `qml.Hamiltonian` points to either old Hamiltonian or
LinearCombination depending on `__use_new_opmath`
- [x] support Hermitian
- [ ] matrix method (optional)
- [ ] adjoint method (optional)
- [ ] batching support (optional)
- [ ] qml.dot points to LinearCombination
- [ ] take care of xfails (mostly about raiseing errors in other parts
of the codebase)
- [ ] update logic of adjoint differentiation to catch attempt to
differentiate lincomb coeffs
- [x] support in qml.equal
- [x]  add special matmul with other product or non-composite ops
- [x] add special matmul with other LinearCombination


[sc-56704]

---------

Co-authored-by: lillian542 <38584660+lillian542@users.noreply.github.com>
Co-authored-by: Astral Cai <astral.cai@xanadu.ai>
Co-authored-by: Mudit Pandey <mudit.pandey@xanadu.ai>
Co-authored-by: Christina Lee <christina@xanadu.ai>
Co-authored-by: Pietropaolo Frisoni <pietropaolo.frisoni@xanadu.ai>
Co-authored-by: albi3ro <chrissie.c.l@gmail.com>
Co-authored-by: Utkarsh <utkarshazad98@gmail.com>
Co-authored-by: lillian542 <Lillian.frederiksen@xanadu.ai>
Co-authored-by: Alex Preciado <alex.preciado@xanadu.ai>
Co-authored-by: Thomas R. Bromley <49409390+trbromley@users.noreply.github.com>
Co-authored-by: Vincent Michaud-Rioux <vincentm@nanoacademic.com>
Co-authored-by: Josh Izaac <josh146@gmail.com>
Co-authored-by: Nathan Killoran <co9olguy@users.noreply.github.com>
Co-authored-by: Matthew Silverman <matthews@xanadu.ai>
Co-authored-by: Mikhail Andrenkov <mikhail@xanadu.ai>
Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>

…nnylane into enable_new_opmath