Repository: PennyLaneAI/pennylane

Issue:
[BUG] compile pre-decomposes at most once
### Expected behavior

I would expect that if I do qml.compile() of a template and its adjoint template, the decompositions would be equivalent but adjoined. However the depth of the decomposition is different:

<img width="300" alt="Captura de pantalla 2023-11-07 a las 13 58 38" src="https://github.com/PennyLaneAI/pennylane/assets/65235481/b6d7525e-e751-47ea-824e-d1bda5701f73">

<img width="300" alt="Captura de pantalla 2023-11-07 a las 13 58 49" src="https://github.com/PennyLaneAI/pennylane/assets/65235481/967a1e09-0203-4501-b0c2-3c2e345cc5d7">

As we can see in one case it expands the QFT and in another it does not. I would have liked that they were equivalent so in both cases it shows (QFT or QFT^t)



### Actual behavior

Explained above

### Additional information

_No response_

### Source code

```shell
import pennylane as qml

dev = qml.device("default.qubit")

@qml.qnode(dev)
@qml.compile()
def circuit():
  #qml.QuantumPhaseEstimation(qml.PauliZ(0), estimation_wires = range(1,3))
  qml.adjoint(qml.QuantumPhaseEstimation)(qml.PauliZ(0), estimation_wires = range(1,3))
  return qml.state()

qml.draw_mpl(circuit)()
```


### Tracebacks

_No response_

### System information

```shell
Name: PennyLane
Version: 0.33.0
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, typing-extensions
Required-by: PennyLane-Lightning

Platform info:           Linux-5.15.120+-x86_64-with-glibc2.35
Python version:          3.10.12
Numpy version:           1.23.5
Scipy version:           1.11.3
Installed devices:
```


### Existing GitHub issues

- [X] I have searched existing GitHub issues to make sure the issue does not already exist.


Discussion:

Resolving patch:
diff --git a/pennylane/transforms/compile.py b/pennylane/transforms/compile.py
--- a/pennylane/transforms/compile.py
+++ b/pennylane/transforms/compile.py
@@ -52,14 +52,15 @@ def compile(
             tape and/or quantum function transforms to apply.
         basis_set (list[str]): A list of basis gates. When expanding the tape,
             expansion will continue until gates in the specific set are
-            reached. If no basis set is specified, no expansion will be done.
+            reached. If no basis set is specified, a default of
+            ``pennylane.ops.__all__`` will be used. This decomposes templates and
+            operator arithmetic.
         num_passes (int): The number of times to apply the set of transforms in
             ``pipeline``. The default is to perform each transform once;
             however, doing so may produce a new circuit where applying the set
             of transforms again may yield further improvement, so the number of
             such passes can be adjusted.
-        expand_depth (int): When ``basis_set`` is specified, the depth to use
-            for tape expansion into the basis gates.
+        expand_depth (int): The depth to use for tape expansion into the basis gates.
 
     Returns:
         qnode (QNode) or quantum function (Callable) or tuple[List[QuantumTape], function]: The compiled circuit. The output type is explained in :func:`qml.transform <pennylane.transform>`.
@@ -181,17 +182,12 @@ def qfunc(x, y, z):
     # don't queue anything as a result of the expansion or transform pipeline
 
     with QueuingManager.stop_recording():
-        if basis_set is not None:
-            expanded_tape = tape.expand(
-                depth=expand_depth, stop_at=lambda obj: obj.name in basis_set
-            )
-        else:
-            # Expands out anything that is not a single operation (i.e., the templates)
-            # expand barriers when `only_visual=True`
-            def stop_at(obj):
-                return (obj.name in all_ops) and (not getattr(obj, "only_visual", False))
-
-            expanded_tape = tape.expand(stop_at=stop_at)
+        basis_set = basis_set or all_ops
+
+        def stop_at(obj):
+            return obj.name in basis_set and (not getattr(obj, "only_visual", False))
+
+        expanded_tape = tape.expand(depth=expand_depth, stop_at=stop_at)
 
         # Apply the full set of compilation transforms num_passes times
         for _ in range(num_passes):


Test patch:
diff --git a/tests/transforms/test_compile.py b/tests/transforms/test_compile.py
--- a/tests/transforms/test_compile.py
+++ b/tests/transforms/test_compile.py
@@ -302,9 +302,12 @@ def test_compile_template(self):
 
         # Push commuting gates to the right and merging rotations gives a circuit
         # with alternating RX and CNOT gates
+        # pylint: disable=expression-not-assigned
         def qfunc(x, params):
             qml.templates.AngleEmbedding(x, wires=range(3))
-            qml.templates.BasicEntanglerLayers(params, wires=range(3))
+            qml.adjoint(
+                qml.adjoint(qml.templates.BasicEntanglerLayers(params, wires=range(3)))
+            ) ** 2
             return qml.expval(qml.PauliZ(wires=2))
 
         dev = qml.device("default.qubit", wires=3)
@@ -321,7 +324,7 @@ def qfunc(x, params):
         transformed_result = transformed_qnode(x, params)
         assert np.allclose(original_result, transformed_result)
 
-        names_expected = ["RX", "CNOT"] * 6
+        names_expected = ["RX", "CNOT"] * 12
         wires_expected = [
             Wires(0),
             Wires([0, 1]),
@@ -329,7 +332,7 @@ def qfunc(x, params):
             Wires([1, 2]),
             Wires(2),
             Wires([2, 0]),
-        ] * 2
+        ] * 4
 
         compare_operation_lists(transformed_qnode.qtape.operations, names_expected, wires_expected)