Incorrect elements in anti commutation graph for abelian grouping

[irajput]

Information

• Qiskit Terra version: 0.17.4
• Python version: 3.9.5
• Operating system: macOS Big Sur

What is the current behavior?

The _anti_commutation_graph function which is used by group_ops in AbelianGrouper sometimes returns anti commutation graphs which has edges with commutable operators instead of anti-commutable operators.

Below, acg is the anti commutation graph for sparse_pauli. acg contains (1,2) which corresponds to the Sparse Pauli operators ('XX', (coeff)) and ('YY', (coeff)) which are commutable. Therefore, (1,2) should not be in the anti commutation graph.

Steps to reproduce the problem

from qiskit.quantum_info import SparsePauliOp, Operator
from qiskit.opflow import PauliSumOp, AbelianGrouper
import numpy as np
sparse_pauli=SparsePauliOp([[0, 0, 0, 0],[1,1,0,0],[1,1,1,1]])
#sparse_pauli.to_list() returns [('II', (1+0j)), ('XX', (1+0j)), ('YY', (1+0j))]
pauli_sum=PauliSumOp(sparse_pauli)
acg=AbelianGrouper()._anti_commutation_graph(pauli_sum)

What is the expected behavior?

The _anti_commutation_graph function should only return edges which have operators that are not commutable.

Note: Fixing #6624 would also fix this issue for group_ops implemented with PauliLists.

[jlapeyre]

I think it may not be worth fixing this bug. I don't know the origin anyway. It depends on how when the PauliList PR is merged (#5993) and when a PR for #6624 might be ready.

[ikkoham]

This is not a bug. AbelianGrouper assumes qubit-wise commutativity, so XX and YY can't belong the same group.

[jlapeyre]

What does qubit-wise commutativity mean? Each pair of qubits must commute? Doesn't this mean only identical strings commute?
EDIT: Ok. Isha showed me an explanation.

[ikkoham]

Each pair of qubits must commute?

Yes. for example, XI and XX are qubit-wise commutable.
This concept is introduced in https://arxiv.org/abs/1704.05018.
They said grouping into TPB sets.

[jlapeyre]

What is the meaning of TPB? I see this acronym sometimes in qiskit comments, for example in the abelian_grouper.py.
EDIT: TPB = Tensor Product Basis.