FermionicOp.simplify does not document its cross-behavior with FermionicOp.index_order

[AlexandreFoley]

Environment

• Qiskit Nature version: qiskit-nature==0.5.2
• Python version: 3.10.10
• Operating system: macos

What is happening?

Calling simplify on a FermionicOp doesn't yield a fully simplified representation of the operator.
Calling normal order first seems to be a work around for this issue, but it makes simplifying SparseLabelOp in a generic manner impossible.

Simplify for Fermionic op should likely make use of normal ordering to perform simplification. The current implementation doesn't work.
This is similar to issue #901, but it's not that simplify is not agressive enough, but that it fail entirely to simplify the fermionic operator.

How can we reproduce the issue?

from qiskit_nature.second_q.hamiltonians import FermiHubbardModel
from qiskit_nature.second_q.hamiltonians.lattices import LineLattice,BoundaryCondition
from qiskit_nature.second_q.operators import commutators
from qiskit_nature.second_q.operators import FermionicOp

#Create a hamiltonian, Hubbard model because that's what i have on hand.
U = 4
mu = 2
t=1
L=2
line_lattice = LineLattice(num_nodes=L, boundary_condition=BoundaryCondition.OPEN)
Hamiltonian = FermiHubbardModel(line_lattice.uniform_parameters(uniform_interaction=t,uniform_onsite_potential=-mu),onsite_interaction=U).second_q_op()

#secondary operator
C0 = FermionicOp({ "+_0": 1 },    num_spin_orbitals=4)

#create a more complicated fermionic operator
F = commutators.commutator(Hamiltonian,commutators.commutator(Hamiltonian,C0))
print(len(F)) #prints 106
print(len(F.simplify())) #prints 106
print(len(F.normal_order())) #prints 6

What should happen?

simplify should manage to reduce the number of label in F as much or more than normal_order

Any suggestions?

aplying normal ordering could be part of simplify's implementation.

[woodsp-ibm]

Had you seen this https://qiskit.org/ecosystem/nature/stubs/qiskit_nature.second_q.operators.FermionicOp.index_order.html It was done as part of #902 which addressed the #901 issue you referred to above. #902 has some comment in there as regards the choice things that way.

[AlexandreFoley]

It's of little help when working with generic SparseLabelOp. simplify is apart of the generic interface, but not normal_order, nor index_order.
At best, those are workaround for FermionicOp simplify failing to simplify the fermionic operator.

[mrossinek]

With the current development version of Qiskit Nature F has only 6 terms from the beginning (and never 106).
Regardless of that, as Steve also pointed out earlier, this behavior is intentional as explained here:

qiskit-nature/qiskit_nature/second_q/operators/fermionic_op.py

Lines 437 to 444 in eb30b6b

	.. note::
	You can use this method to achieve the most aggressive simplification of an operator
	without changing the operation order per index. :meth:`simplify` does *not* reorder the
	terms and, thus, cannot deduce ``-_0 +_1`` and ``+_1 -_0 +_0 -_0`` to be
	identical labels. Calling this method will reorder the latter label to
	``-_0 +_0 -_0 +_1``, after which :meth:`simplify` will be able to correctly collapse
	these two labels into one.

Arguably, we should add this note to the docstring of the FermionicOp.simplify method, too.