Repository: PennyLaneAI/pennylane

Issue:
[BUG] op_math adding two GellMann observables doesn't treat them as different Hermitians
### Expected behavior

```
>>> gell_1 = qml.GellMann(wires=0, index=3)
>>> gell_2 = qml.GellMann(wires=0, index=8)
>>> ham_added = gell_1 + gell_2
>>> print(type(ham_added))
<class 'pennylane.ops.qubit.hamiltonian.Hamiltonian'>
>>> print(ham_added.coeffs)
array([1,1])
>>> print(ham_added.ops)
[GellMann3(wires=[0]), GellMann8(wires=[0])]
```

The two GellMann observables should be appended in a Hamiltonian object as shown.

### Actual behavior

```
>>> gell_1 = qml.GellMann(wires=0, index=3)
>>> gell_2 = qml.GellMann(wires=0, index=8)
>>> ham_added = gell_1 + gell_2
>>> print(type(ham_added))
<class 'pennylane.ops.qubit.hamiltonian.Hamiltonian'>
>>> print(ham_added.coeffs)
array([2])
>>> print(ham_added.ops)
[GellMann3(wires=[0])]
```
Currently when adding two GellMann observables Pennylane interprets them as the same type and concatinates them together taking the first as it's type.

### Additional information

By using ```qml.operation.enable_new_opmath()``` the bahaviour seems to work correctly.

```
>>> qml.operation.enable_new_opmath()

>>> gell_1 = qml.GellMann(wires=0, index=3)
>>> gell_2 = qml.GellMann(wires=0, index=8)
>>> ham_added = gell_1 + gell_2
>>> print(type(ham_added))
<class 'pennylane.ops.op_math.sum.Sum'>
>>> print(ham_added)
GellMann3(wires=[0]) + GellMann8(wires=[0])
```

Similar to #3114, on comment chain I was told to make a new issue

### Source code

_No response_

### Tracebacks

_No response_

### System information

```shell
Name: PennyLane
Version: 0.32.0.dev0
Summary: PennyLane is a Python quantum machine learning library by Xanadu Inc.
Home-page: https://github.com/PennyLaneAI/pennylane
Author: 
Author-email: 
License: Apache License 2.0
Location: /usr/local/lib/python3.10/dist-packages
Requires: appdirs, autograd, autoray, cachetools, networkx, numpy, pennylane-lightning, requests, rustworkx, scipy, semantic-version, toml
Required-by: PennyLane-Lightning

Platform info:           Linux-5.15.107+-x86_64-with-glibc2.31
Python version:          3.10.12
Numpy version:           1.22.4
Scipy version:           1.10.0
Installed devices:
- default.gaussian (PennyLane-0.32.0.dev0)
- default.mixed (PennyLane-0.32.0.dev0)
- default.qubit (PennyLane-0.32.0.dev0)
- default.qubit.autograd (PennyLane-0.32.0.dev0)
- default.qubit.jax (PennyLane-0.32.0.dev0)
- default.qubit.tf (PennyLane-0.32.0.dev0)
- default.qubit.torch (PennyLane-0.32.0.dev0)
- default.qutrit (PennyLane-0.32.0.dev0)
- null.qubit (PennyLane-0.32.0.dev0)
- lightning.qubit (PennyLane-Lightning-0.31.0)
```


### Existing GitHub issues

- [X] I have searched existing GitHub issues to make sure the issue does not already exist.


Discussion:
Thanks for opening this issue @Gabriel-Bottrill .

We could potentially override the dunder methods on qutrit observables so they always use operator arithmetic.

Resolving patch:
diff --git a/pennylane/operation.py b/pennylane/operation.py
--- a/pennylane/operation.py
+++ b/pennylane/operation.py
@@ -1892,6 +1892,8 @@ def _obs_data(self):
 
         for ob in obs:
             parameters = tuple(param.tobytes() for param in ob.parameters)
+            if isinstance(ob, qml.GellMann):
+                parameters += (ob.hyperparameters["index"],)
             tensor.add((ob.name, ob.wires, parameters))
 
         return tensor
diff --git a/pennylane/ops/qubit/hamiltonian.py b/pennylane/ops/qubit/hamiltonian.py
--- a/pennylane/ops/qubit/hamiltonian.py
+++ b/pennylane/ops/qubit/hamiltonian.py
@@ -564,6 +564,8 @@ def _obs_data(self):
                 parameters = tuple(
                     str(param) for param in ob.parameters
                 )  # Converts params into immutable type
+                if isinstance(ob, qml.GellMann):
+                    parameters += (ob.hyperparameters["index"],)
                 tensor.append((ob.name, ob.wires, parameters))
             data.add((co, frozenset(tensor)))
 


Test patch:
diff --git a/tests/ops/qubit/test_hamiltonian.py b/tests/ops/qubit/test_hamiltonian.py
--- a/tests/ops/qubit/test_hamiltonian.py
+++ b/tests/ops/qubit/test_hamiltonian.py
@@ -786,6 +786,45 @@ def test_data(self):
             (0.5, frozenset([("PauliX", qml.wires.Wires(2), ())])),
         }
 
+    def test_data_gell_mann(self):
+        """Tests that the obs_data method for Hamiltonians with qml.GellMann
+        observables includes the Gell-Mann index."""
+        H = qml.Hamiltonian(
+            [1, -1, 0.5],
+            [
+                qml.GellMann(wires=0, index=3),
+                qml.GellMann(wires=0, index=3) @ qml.GellMann(wires=1, index=1),
+                qml.GellMann(wires=2, index=2),
+            ],
+        )
+        data = H._obs_data()
+
+        assert data == {
+            (1, frozenset([("GellMann", qml.wires.Wires(0), (3,))])),
+            (
+                -1,
+                frozenset(
+                    [("GellMann", qml.wires.Wires(0), (3,)), ("GellMann", qml.wires.Wires(1), (1,))]
+                ),
+            ),
+            (0.5, frozenset([("GellMann", qml.wires.Wires(2), (2,))])),
+        }
+
+    def test_compare_gell_mann(self):
+        """Tests that the compare method returns the correct result for Hamiltonians
+        with qml.GellMann present."""
+        H1 = qml.Hamiltonian([1], [qml.GellMann(wires=2, index=2)])
+        H2 = qml.Hamiltonian([1], [qml.GellMann(wires=2, index=1) @ qml.GellMann(wires=1, index=2)])
+        H3 = qml.Hamiltonian([1], [qml.GellMann(wires=2, index=1)])
+        H4 = qml.Hamiltonian([1], [qml.GellMann(wires=2, index=1) @ qml.GellMann(wires=1, index=3)])
+
+        assert H1.compare(qml.GellMann(wires=2, index=2)) is True
+        assert H1.compare(qml.GellMann(wires=2, index=1)) is False
+        assert H1.compare(H3) is False
+        assert H2.compare(qml.GellMann(wires=2, index=1) @ qml.GellMann(wires=1, index=2)) is True
+        assert H2.compare(qml.GellMann(wires=2, index=2) @ qml.GellMann(wires=1, index=2)) is False
+        assert H2.compare(H4) is False
+
     def test_hamiltonian_equal_error(self):
         """Tests that the correct error is raised when compare() is called on invalid type"""
 
diff --git a/tests/ops/qutrit/test_qutrit_observables.py b/tests/ops/qutrit/test_qutrit_observables.py
--- a/tests/ops/qutrit/test_qutrit_observables.py
+++ b/tests/ops/qutrit/test_qutrit_observables.py
@@ -371,6 +371,18 @@ def test_matrix(self, index, mat, eigs, tol):
         assert np.allclose(res_static, mat)
         assert np.allclose(res_dynamic, mat)
 
+    def test_obs_data(self):
+        """Test that the _obs_data() method of qml.GellMann returns the correct
+        observable data."""
+        ob1 = qml.GellMann(wires=0, index=2)
+        ob2 = qml.GellMann(wires=0, index=2) @ qml.GellMann(wires=1, index=1)
+
+        assert ob1._obs_data() == {("GellMann", qml.wires.Wires(0), (2,))}
+        assert ob2._obs_data() == {
+            ("GellMann", qml.wires.Wires(0), (2,)),
+            ("GellMann", qml.wires.Wires(1), (1,)),
+        }
+
     @pytest.mark.parametrize("index, mat, eigs", GM_OBSERVABLES)
     def test_eigvals(self, index, mat, eigs, tol):
         """Test that the Gell-Mann eigenvalues are correct"""