Deprecate running pulse jobs on fake backends

With Aer 0.13 removing the capability, this is not something we will be
able to support moving forwards.  This deprecates the functionality, in
preparation for its full removal in Qiskit 1.0.

qiskit/providers/fake_provider/fake_backend.py

@@ -553,6 +553,14 @@ def run(self, run_input, **kwargs):
                 "QuantumCircuit, Schedule, or a list of either" % circuits
             )
         if pulse_job:
+            warnings.warn(
+                "Simulating pulse jobs on fake backends is deprecated as of Qiskit 0.46,"
+                " and the functionality will be removed in Qiskit 1.0, due to Qiskit Aer removing"
+                " its pulse-simulation capabilities in its most recent versions."
+                " You can try using Qiskit Dynamics instead.",
+                DeprecationWarning,
+                stacklevel=2,
+            )
             if _optionals.HAS_AER:
                 import qiskit_aer
                 from qiskit_aer.pulse import PulseSystemModel

releasenotes/notes/deprecate-aer-hooks-1bc00a96ebb269f6.yaml

@@ -7,3 +7,8 @@ deprecations:
   - |
     Importing from ``qiskit.providers.aer`` is deprecated and will stop working in Qiskit 1.0.  You
     should instead import from ``qiskit_aer``, which is a drop-in replacement.
+  - |
+    Running pulse jobs on backends from :mod:`qiskit.providers.fake_provider` is deprecated, and
+    all support will be removed in Qiskit 1.0.  This is due to Qiskit Aer removing its simulation
+    functionality for such jobs.  For low-level Hamiltonain-simulation workloads, consider using
+    a specialised library such as `Qiskit Dynamics <https://qiskit.org/ecosystem/dynamics>`__.