fix + test for controlling circuits with nontrivial phase

qiskit/transpiler/passes/synthesis/hls_plugins.py

@@ -425,6 +425,7 @@
     HalfAdderGate,
     FullAdderGate,
     MultiplierGate,
+    GlobalPhaseGate,
 )
 from qiskit.circuit.annotated_operation import (
     AnnotatedOperation,
@@ -1727,6 +1728,15 @@ def _apply_annotations(circuit: QuantumCircuit, modifiers: list[Modifier]) -> Qu
 
             # Apply the control modifier to each gate in the circuit.
             controlled_circuit = QuantumCircuit(modifier.num_ctrl_qubits + circuit.num_qubits)
+            if circuit.global_phase != 0:
+                controlled_op = GlobalPhaseGate(circuit.global_phase).control(
+                    num_ctrl_qubits=modifier.num_ctrl_qubits,
+                    label=None,
+                    ctrl_state=modifier.ctrl_state,
+                    annotated=False,
+                )
+                controlled_qubits = list(range(0, modifier.num_ctrl_qubits))
+                controlled_circuit.append(controlled_op, controlled_qubits)
             for inst in circuit:
                 inst_op = inst.operation
                 inst_qubits = inst.qubits

test/python/transpiler/test_high_level_synthesis.py

@@ -1617,6 +1617,24 @@ def test_annotated_circuit(self):
         qct = pass_(qc)
         self.assertEqual(Statevector(qc), Statevector(qct))
 
+    def test_annotated_circuit_with_phase(self):
+        inner = QuantumCircuit(2)
+        inner.global_phase = 1
+        inner.h(0)
+        inner.cx(0, 1)
+        gate = inner.to_gate()
+
+        qc1 = QuantumCircuit(3)
+        qc1.append(gate.control(annotated=False), [0, 1, 2])
+        qct1 = HighLevelSynthesis(basis_gates=["cx", "u"])(qc1)
+
+        qc2 = QuantumCircuit(3)
+        qc2.append(gate.control(annotated=True), [0, 1, 2])
+        qct2 = HighLevelSynthesis(basis_gates=["cx", "u"])(qc2)
+
+        self.assertEqual(Operator(qc1), Operator(qc2))
+        self.assertEqual(Operator(qct1), Operator(qct2))
+
     def test_annotated_rec(self):
         """Test synthesis with annotated custom gates and recursion."""
         inner2 = QuantumCircuit(2)