Fix regression in scars tutorial: properly use noiseless backend when NO_NOISE=True (#2982)

* Fix regression in scars tutorial: properly use noiseless backend when NO_NOISE=True

The ibmq_executor_full function was always using the noisy backend,
ignoring the NO_NOISE flag. When NO_NOISE=True, the ideal Trotter
curve should be computed on backend_noiseless, not backend.

Also add warnings.filterwarnings to suppress UserWarnings that appear
during execution on the stable docs page.

Fixes #2973

* Simplify: use exec_backend instead of inline conditional

Resolves review comment by natestemen

Author: hijirii

docs/source/examples/quantum_simulation_scars_ibmq.md

@@ -14,6 +14,11 @@ kernelspec:
 ```{tags} qiskit, zne, advanced
 ```
 
+```{code-cell} ipython3
+import warnings
+warnings.filterwarnings("ignore", category=UserWarning)
+```
+
 ```{code-cell} ipython3
 import qiskit
 from qiskit import QuantumCircuit
@@ -188,9 +193,12 @@ def ibmq_executor_full(circuit: qiskit.QuantumCircuit,
         shots: Number of times to execute the circuit to compute 
         the expectation value.
     """
+    # Select the appropriate backend based on noise flag
+    exec_backend = backend_noiseless if NO_NOISE else backend
+    
     # Transpile the circuit so it can be properly run
     pm = generate_preset_pass_manager(
-        backend=backend, 
+        backend=exec_backend, 
         optimization_level=0, 
         basis_gates=noise_model.basis_gates if noise_model else None,
     )
@@ -200,8 +208,8 @@ def ibmq_executor_full(circuit: qiskit.QuantumCircuit,
     if not isinstance(exec_circuit, list):
         exec_circuit = [exec_circuit]
 
-    # Run the circuit
-    sampler = Sampler(backend)
+    # Run the circuit on the selected backend
+    sampler = Sampler(exec_backend)
     job = sampler.run(exec_circuit, shots=shots)
 
     # Convert from raw measurement counts to the expectation value