Repository: PennyLaneAI/pennylane

Issue:
[BUG] Wrong expectation value of Hamiltonian with return qml.expval(H)
### Expected behavior

Expected to get the same expectation value of a qml.Hamiltonian no matter
1. how the operators and  the corresponding coefficient are ordered when it was constructed.
2. applied the grouping method ('QWC') or not.

### Actual behavior

When the below conditions are satisfied, the expectation value will not take the value (coefficient) of qml.Identity into account.
1. There is a "grouping_type" argument when qml.Hamiltonian is constructed.
2. The operator list is not  in the order of qubit number.  

Example
`H_disordered = qml.Hamiltonian([1.0, 1.0, 1.0], [qml.PauliZ(1), qml.PauliZ(0), qml.Identity(0)], grouping_type='QWC')`
The list of operator is disordered in terms of qubit number.
However its expectation value differs with the value of qml.Identity from an ordered Hamiltonian  
`H_ordered = qml.Hamiltonian([1.0, 1.0, 1.0], [qml.Identity(0), qml.PauliZ(0), qml.PauliZ(1)], grouping_type='QWC')`


### Additional information

_No response_

### Source code

```shell
import pennylane as qml

dev = qml.device("default.qubit", wires=2, shots=1000)
@qml.qnode(dev)
def circuit():
    return qml.expval(H)

H = qml.Hamiltonian([1.0, 1.0, 1.0], [qml.PauliZ(0), qml.PauliZ(1), qml.Identity(0)], grouping_type='QWC')
print(circuit())

H = qml.Hamiltonian([1.0, 1.0, 1.0], [qml.PauliZ(1), qml.PauliZ(0), qml.Identity(0)], grouping_type='QWC')
print(circuit())

H = qml.Hamiltonian([1.0, 1.0, 1.0], [qml.PauliZ(1), qml.PauliZ(0), qml.Identity(0)], grouping_type='commuting')
print(circuit())

H = qml.Hamiltonian([1.0, 1.0, 1.0], [qml.PauliZ(1), qml.PauliZ(0), qml.Identity(0)])
print(circuit())
```


### Tracebacks

_No response_

### System information

```shell
Platform info:           macOS-13.0.1-arm64-arm-64bit
Python version:          3.9.12
Numpy version:           1.23.3
Scipy version:           1.9.3
Installed devices:
- default.gaussian (PennyLane-0.30.0)
- default.mixed (PennyLane-0.30.0)
- default.qubit (PennyLane-0.30.0)
- default.qubit.autograd (PennyLane-0.30.0)
- default.qubit.jax (PennyLane-0.30.0)
- default.qubit.tf (PennyLane-0.30.0)
- default.qubit.torch (PennyLane-0.30.0)
- default.qutrit (PennyLane-0.30.0)
- null.qubit (PennyLane-0.30.0)
- lightning.qubit (PennyLane-Lightning-0.30.0)
```


### Existing GitHub issues

- [X] I have searched existing GitHub issues to make sure the issue does not already exist.

Discussion:
albi3ro: > very clean fix, nice! I can't help but ask about the original concern you mention in the PR description when going Identity -> binary_repr -> Identity, which will still occur after this fix. The first step is lossy, so I'm good to have this unexpected behaviour because what else can you do. However, it makes me wonder if we're missing little bugs like this elsewhere. I suppose it's hard to know until we hit them. Happy to approve regardless, but something to consider

Once we port all these "utilities" over to relying on `PauliWord` and `PauliSentence` instead, we won't have to deal with the binary rep quirks anymore, so I'm not too concerned about the longevity of this problem.

Patch:
diff --git a/doc/releases/changelog-dev.md b/doc/releases/changelog-dev.md
index 36cd9cb70..3018521fe 100644
--- a/doc/releases/changelog-dev.md
+++ b/doc/releases/changelog-dev.md
@@ -178,6 +178,10 @@
 * Removes a patch in `interfaces/autograd.py` that checks for the `strawberryfields.gbs` device.  That device
   is pinned to PennyLane <= v0.29.0, so that patch is no longer necessary.
 
+* `qml.pauli.are_identical_pauli_words` now treats all identities as equal. Identity terms on Hamiltonians with non-standard
+  wire orders are no longer eliminated.
+  [(#4161)](https://github.com/PennyLaneAI/pennylane/pull/4161)
+
 <h3>Contributors ✍️</h3>
 
 This release contains contributions from (in alphabetical order):
diff --git a/pennylane/pauli/utils.py b/pennylane/pauli/utils.py
index 2e7e82f57..fdfc72982 100644
--- a/pennylane/pauli/utils.py
+++ b/pennylane/pauli/utils.py
@@ -148,6 +148,9 @@ def are_identical_pauli_words(pauli_1, pauli_2):
     pauli_1 = getattr(pauli_1, "prune", lambda: pauli_1)()
     pauli_2 = getattr(pauli_2, "prune", lambda: pauli_2)()
 
+    if isinstance(pauli_1, qml.Identity) and isinstance(pauli_2, qml.Identity):
+        return True
+
     if isinstance(pauli_1, paulis_with_identity) and isinstance(pauli_2, paulis_with_identity):
         return (pauli_1.wires, pauli_1.name) == (pauli_2.wires, pauli_2.name)
 


Test patch:
diff --git a/tests/ops/qubit/test_hamiltonian.py b/tests/ops/qubit/test_hamiltonian.py
index 8bc9fe1c6..dea965a4f 100644
--- a/tests/ops/qubit/test_hamiltonian.py
+++ b/tests/ops/qubit/test_hamiltonian.py
@@ -1479,6 +1479,14 @@ class TestHamiltonianArithmeticJax:
 class TestGrouping:
     """Tests for the grouping functionality"""
 
+    def test_indentities_preserved(self):
+        """Tests that the grouping indices do not drop identity terms when the wire order is nonstandard."""
+
+        obs = [qml.PauliZ(1), qml.PauliZ(0), qml.Identity(0)]
+
+        H = qml.Hamiltonian([1.0, 1.0, 1.0], obs, grouping_type="qwc")
+        assert H.grouping_indices == [[0, 1, 2]]
+
     def test_grouping_is_correct_kwarg(self):
         """Basic test checking that grouping with a kwarg works as expected"""
         a = qml.PauliX(0)
diff --git a/tests/pauli/test_pauli_utils.py b/tests/pauli/test_pauli_utils.py
index 60fdbba33..dacde9300 100644
--- a/tests/pauli/test_pauli_utils.py
+++ b/tests/pauli/test_pauli_utils.py
@@ -309,6 +309,11 @@ class TestGroupingUtils:
         assert not are_identical_pauli_words(pauli_word_1, pauli_word_4)
         assert not are_identical_pauli_words(pauli_word_3, pauli_word_4)
 
+    def test_identities_always_pauli_words(self):
+        """Tests that identity terms are always identical."""
+        assert are_identical_pauli_words(qml.Identity(0), qml.Identity("a"))
+        assert are_identical_pauli_words(qml.Identity((-2, -3)), qml.Identity("wire"))
+
     @pytest.mark.parametrize("non_pauli_word", non_pauli_words)
     def test_are_identical_pauli_words_non_pauli_word_catch(self, non_pauli_word):
         """Tests TypeError raise for when non-Pauli word Pennylane operators/operations are given