Fix tutorial_pulse_programming101.py to type cast expval (#1317)

**Context:**

In PennyLaneAI/pennylane#6939, a fix was made to
`qml.expval` so that it no longer silently converts the result to a real
number. However, in `tutorial_pulse_programming101.py` we use a
Hamiltonian with complex-typed coefficients. This resulted in an
imaginary expectation value (`X+0j`) and broke the `jax.value_and_grad`
function.

**Description of change:**

Wrap the `qnode` function to get the expectation value and convert it to
a real number. This allows `jax.value_and_grad` to be used and fixes the
demo.

demonstrations/tutorial_pulse_programming101.metadata.json

@@ -6,7 +6,7 @@
         }
     ],
     "dateOfPublication": "2023-03-08T00:00:00+00:00",
-    "dateOfLastModification": "2024-10-07T00:00:00+00:00",
+    "dateOfLastModification": "2025-02-14T00:00:00+00:00",
     "categories": [
         "Quantum Hardware",
         "Quantum Computing"

demonstrations/tutorial_pulse_programming101.py

@@ -322,15 +322,21 @@ def wrapped(p, t):
 
 ##############################################################################
 # Now we define the ``qnode`` that computes the expectation value of the molecular Hamiltonian.
+# We need to wrap the ``qnode`` in a function so that we can convert the expectation value to a real number.
+# This will enable use to make use of gradient descent methods that require real-valued loss functions.
 
 dev = qml.device("default.qubit", wires=range(n_wires))
 
-@qml.qnode(dev, interface="jax")
 def qnode(theta, t=duration):
-    qml.BasisState(jnp.array(data.tapered_hf_state), wires=H_obj.wires)
-    qml.evolve(H_pulse)(params=(*theta, *theta), t=t)
-    return qml.expval(H_obj)
 
+    @qml.qnode(dev)
+    def _qnode_inner(theta, t=duration):
+        qml.BasisState(jnp.array(data.tapered_hf_state), wires=H_obj.wires)
+        qml.evolve(H_pulse)(params=(*theta, *theta), t=t)
+        return qml.expval(H_obj)
+
+    expectation_value = _qnode_inner(theta, t)  # Execute the qnode
+    return jnp.real(expectation_value)  # Typecast to real number
 
 value_and_grad = jax.jit(jax.value_and_grad(qnode))