Treat classically controlled ops as non-Clifford in dynamical_decoupling

cirq-core/cirq/ops/classically_controlled_operation.py

@@ -125,6 +125,9 @@ def with_qubits(self, *new_qubits) -> cirq.Operation:
             *self._conditions
         )
 
+    def _has_unitary_(self) -> bool:
+        return False
+
     def _decompose_with_context_(self, *, context: cirq.DecompositionContext):
         result = protocols.decompose_once(self._sub_operation, None, flatten=False, context=context)
         if result is None:

cirq-core/cirq/transformers/dynamical_decoupling_test.py

@@ -20,7 +20,7 @@
 import pytest
 
 import cirq
-from cirq import add_dynamical_decoupling, CNOT, CZ, CZPowGate, H, X, Y, Z
+from cirq import add_dynamical_decoupling, CNOT, CZ, CZPowGate, H, I, measure, X, Y, Z
 
 
 def assert_sim_eq(circuit1: cirq.AbstractCircuit, circuit2: cirq.AbstractCircuit):
@@ -53,6 +53,22 @@ def assert_dd(
     assert_sim_eq(input_circuit, transformed_circuit)
 
 
+def test_classically_controlled_no_update_succeeds():
+    """Test case diagrams.
+    Input:
+    a: ───M───I───
+          ║   ║
+    a: ═══@═══^═══
+    """
+    a = cirq.NamedQubit('a')
+
+    add_dynamical_decoupling(
+        cirq.Circuit(
+            cirq.Moment(measure(a, key="a")), cirq.Moment(I(a).with_classical_controls("a"))
+        )
+    )
+
+
 def test_no_insertion():
     """Test case diagrams.
     Input: