Repository: PennyLaneAI/pennylane (python)
Issue: Issue when using batched data on StronglyEntanglingLayers template.[BUG]
### Expected behavior

I want to use the StronglyEntanglingLayers template to process batched data. the shape of the data is like [batch_size, layers, wires, 3]. 

### Actual behavior

Some errors happened: 
```
.../pennylane/templates/layers/strongly_entangling.py", line 209, in compute_decomposition
    weights[..., l, i, 0],
IndexError: index 10 is out of bounds for dimension 1 with size 10
```

### Additional information

I check the strongly_entangling.py and find this bug may comes from this code:
```
201 line:  n_layers = qml.math.shape(weights)[0]
```

the correct code may be 

```
201 line:  n_layers = qml.math.shape(weights)[-3]
```

### Source code

_No response_

### Tracebacks

_No response_

### System information

```shell
I run pennylane on the server, and the bug I think is not related to the system.
```


### Existing GitHub issues

- [X] I have searched existing GitHub issues to make sure the issue does not already exist.
Pull request description: Fix decomposition of `StronglyEntanglingLayers` with broadcasting
**Context:**
The decomposition of `StronglyEntanglingLayers` "almost" takes broadcasting into account: The indexing into the `weights` tensor is fine, but the number of layers is not determined correctly, leading to flawed indexing, and in particular to #5695.

**Description of the Change:**
Fix the `n_layers` determined from `weights` in `compute_decomposition`.

**Benefits:**

**Possible Drawbacks:**

**Related GitHub Issues:**
Fixes #5695 

[sc-63457]

Patch:
diff --git a/doc/releases/changelog-dev.md b/doc/releases/changelog-dev.md
index bd4e5f0ff..9d88744ff 100644
--- a/doc/releases/changelog-dev.md
+++ b/doc/releases/changelog-dev.md
@@ -144,6 +144,9 @@
 
 <h3>Bug fixes 🐛</h3>
 
+* The decomposition of `StronglyEntanglingLayers` is now compatible with broadcasting.
+  [(#5716)](https://github.com/PennyLaneAI/pennylane/pull/5716)
+
 * `qml.cond` can now be applied to `ControlledOp` operations when deferring measurements.
   [(#5725)](https://github.com/PennyLaneAI/pennylane/pull/5725)
 
diff --git a/pennylane/templates/layers/strongly_entangling.py b/pennylane/templates/layers/strongly_entangling.py
index fd07f05ac..554f100e8 100644
--- a/pennylane/templates/layers/strongly_entangling.py
+++ b/pennylane/templates/layers/strongly_entangling.py
@@ -200,7 +200,7 @@ class StronglyEntanglingLayers(Operation):
         CNOT(wires=['a', 'a']),
         CNOT(wires=['b', 'b'])]
         """
-        n_layers = qml.math.shape(weights)[0]
+        n_layers = qml.math.shape(weights)[-3]
         wires = qml.wires.Wires(wires)
         op_list = []
 


Test patch:
diff --git a/tests/templates/test_layers/test_strongly_entangling.py b/tests/templates/test_layers/test_strongly_entangling.py
index d02bbe67e..ed9839b84 100644
--- a/tests/templates/test_layers/test_strongly_entangling.py
+++ b/tests/templates/test_layers/test_strongly_entangling.py
@@ -36,6 +36,7 @@ def test_standard_validity():
     qml.ops.functions.assert_valid(op)
 
 
+@pytest.mark.parametrize("batch_dim", [None, 1, 3])
 class TestDecomposition:
     """Tests that the template defines the correct decomposition."""
 
@@ -57,23 +58,39 @@ class TestDecomposition:
     ]
 
     @pytest.mark.parametrize("n_wires, weight_shape, expected_names, expected_wires", QUEUES)
-    def test_expansion(self, n_wires, weight_shape, expected_names, expected_wires):
+    def test_expansion(self, n_wires, weight_shape, expected_names, expected_wires, batch_dim):
         """Checks the queue for the default settings."""
+        # pylint: disable=too-many-arguments
 
+        if batch_dim is not None:
+            weight_shape = (batch_dim,) + weight_shape
         weights = np.random.random(size=weight_shape)
 
         op = qml.StronglyEntanglingLayers(weights, wires=range(n_wires))
         tape = op.expand()
 
+        if batch_dim is None:
+            param_sets = iter(weights.reshape((-1, 3)))
+        else:
+            param_sets = iter(weights.reshape((batch_dim, -1, 3)).transpose([1, 2, 0]))
+
         for i, gate in enumerate(tape.operations):
             assert gate.name == expected_names[i]
+            if gate.name == "Rot":
+                assert gate.batch_size == batch_dim
+                assert qml.math.allclose(gate.data, next(param_sets))
+            else:
+                assert gate.batch_size is None
             assert gate.wires.labels == tuple(expected_wires[i])
 
     @pytest.mark.parametrize("n_layers, n_wires", [(2, 2), (1, 3), (2, 4)])
-    def test_uses_correct_imprimitive(self, n_layers, n_wires):
+    def test_uses_correct_imprimitive(self, n_layers, n_wires, batch_dim):
         """Test that correct number of entanglers are used in the circuit."""
 
-        weights = np.random.randn(n_layers, n_wires, 3)
+        shape = (n_layers, n_wires, 3)
+        if batch_dim is not None:
+            shape = (batch_dim,) + shape
+        weights = np.random.randn(*shape)
 
         op = qml.StronglyEntanglingLayers(weights=weights, wires=range(n_wires), imprimitive=qml.CZ)
         ops = op.expand().operations
@@ -81,9 +98,10 @@ class TestDecomposition:
         gate_names = [gate.name for gate in ops]
         assert gate_names.count("CZ") == n_wires * n_layers
 
-    def test_custom_wire_labels(self, tol):
+    def test_custom_wire_labels(self, tol, batch_dim):
         """Test that template can deal with non-numeric, nonconsecutive wire labels."""
-        weights = np.random.random(size=(1, 3, 3))
+        shape = (1, 3, 3) if batch_dim is None else (batch_dim, 1, 3, 3)
+        weights = np.random.random(size=shape)
 
         dev = qml.device("default.qubit", wires=3)
         dev2 = qml.device("default.qubit", wires=["z", "a", "k"])
@@ -107,10 +125,13 @@ class TestDecomposition:
     @pytest.mark.parametrize(
         "n_layers, n_wires, ranges", [(2, 2, [1, 1]), (1, 3, [2]), (4, 4, [2, 3, 1, 3])]
     )
-    def test_custom_range_sequence(self, n_layers, n_wires, ranges):
+    def test_custom_range_sequence(self, n_layers, n_wires, ranges, batch_dim):
         """Test that correct sequence of custom ranges are used in the circuit."""
 
-        weights = np.random.randn(n_layers, n_wires, 3)
+        shape = (n_layers, n_wires, 3)
+        if batch_dim is not None:
+            shape = (batch_dim,) + shape
+        weights = np.random.randn(*shape)
 
         op = qml.StronglyEntanglingLayers(weights=weights, wires=range(n_wires), ranges=ranges)
         ops = op.expand().operations