Fix behavior of HLS plugins after preserve_order=False (backport Qiskit#14539) (Qiskit#14547)

* Fix HLS preserve order (Qiskit#14539)

(cherry picked from commit 2982004)

# Conflicts:
#	qiskit/transpiler/passes/synthesis/hls_plugins.py

* Update hls_plugins.py

---------

Co-authored-by: Elena Peña Tapia <[email protected]>

Author: mergify[bot]

qiskit/transpiler/passes/synthesis/hls_plugins.py

@@ -1585,13 +1585,15 @@ def run(self, high_level_object, coupling_map=None, target=None, qubits=None, **
             # Don't do anything if a gate is called "evolution" but is not an
             # actual PauliEvolutionGate
             return None
-
         algo = high_level_object.synthesis
 
+        original_preserve_order = algo.preserve_order
         if "preserve_order" in options and isinstance(algo, ProductFormula):
             algo.preserve_order = options["preserve_order"]
 
-        return algo.synthesize(high_level_object)
+        synth_object = algo.synthesize(high_level_object)
+        algo.preserve_order = original_preserve_order
+        return synth_object
 
 
 class PauliEvolutionSynthesisRustiq(HighLevelSynthesisPlugin):
@@ -1642,6 +1644,7 @@ def run(self, high_level_object, coupling_map=None, target=None, qubits=None, **
             )
             return None
 
+        original_preserve_order = algo.preserve_order
         if "preserve_order" in options:
             algo.preserve_order = options["preserve_order"]
 
@@ -1654,7 +1657,7 @@ def run(self, high_level_object, coupling_map=None, target=None, qubits=None, **
         upto_phase = options.get("upto_phase", False)
         resynth_clifford_method = options.get("resynth_clifford_method", 1)
 
-        return synth_pauli_network_rustiq(
+        synth_object = synth_pauli_network_rustiq(
             num_qubits=num_qubits,
             pauli_network=pauli_network,
             optimize_count=optimize_count,
@@ -1663,3 +1666,5 @@ def run(self, high_level_object, coupling_map=None, target=None, qubits=None, **
             upto_phase=upto_phase,
             resynth_clifford_method=resynth_clifford_method,
         )
+        algo.preserve_order = original_preserve_order
+        return synth_object

releasenotes/notes/fix-preserve-order-0b8c1bf383f42297.yaml

@@ -0,0 +1,6 @@
+---
+fixes:
+  - |
+    Fixed a bug in the :class:`.PauliEvolutionSynthesisDefault` and :class:`.PauliEvolutionSynthesisRustiq` plugins that 
+    modified the `.synthesis` attribute of the original circuit when setting ``preserve_order=False``. The behavior of the
+    plugins has been restored and the original circuit is now preserved throughout the transpilation pipeline.

test/python/transpiler/test_high_level_synthesis.py

@@ -2690,6 +2690,22 @@ def test_trivial_rotations(self, plugin_name):
         self.assertEqual(Operator(qc), Operator(qct))
         self.assertEqual(count_rotation_gates(qct), 1)
 
+    def test_default_preserve_order(self):
+        """Test that option preserve_order is reset."""
+        op = SparsePauliOp(["IIIX", "IIXX", "IYYI", "IIZZ"], coeffs=[1, 2, 3, 4])
+        qc = QuantumCircuit(6)
+        qc.append(PauliEvolutionGate(op), [1, 2, 3, 4])
+        with self.subTest("preserve_order_is_reset"):
+            hls_config = HLSConfig(PauliEvolution=[("default", {"preserve_order": False})])
+            hls_pass = HighLevelSynthesis(hls_config=hls_config)
+            qct = hls_pass(qc)
+            self.assertEqual(qct.depth(), 3)
+            # check that preserve_order is reset and is no longer False
+            hls_config = HLSConfig(PauliEvolution=[("default", {})])
+            hls_pass = HighLevelSynthesis(hls_config=hls_config)
+            qct = hls_pass(qc)
+            self.assertEqual(qct.depth(), 4)
+
     def test_rustiq_upto_options(self):
         """Test non-default Rustiq options upto_phase and upto_clifford."""
         op = SparsePauliOp(["XXXX", "YYYY", "ZZZZ"], coeffs=[1, 2, 3])
@@ -2742,6 +2758,20 @@ def test_rustiq_preserve_order(self):
             cnt_ops = qct.count_ops()
             self.assertEqual(count_rotation_gates(qct), 6)
             self.assertEqual(cnt_ops["cx"], 4)
+        with self.subTest("preserve_order_is_reset"):
+            hls_config = HLSConfig(PauliEvolution=[("rustiq", {"preserve_order": False})])
+            hls_pass = HighLevelSynthesis(hls_config=hls_config)
+            qct = hls_pass(qc)
+            cnt_ops = qct.count_ops()
+            self.assertEqual(count_rotation_gates(qct), 6)
+            self.assertEqual(cnt_ops["cx"], 4)
+            # check that preserve_order is reset and is no longer False
+            hls_config = HLSConfig(PauliEvolution=[("rustiq", {})])
+            hls_pass = HighLevelSynthesis(hls_config=hls_config)
+            qct = hls_pass(qc)
+            cnt_ops = qct.count_ops()
+            self.assertEqual(count_rotation_gates(qct), 6)
+            self.assertEqual(cnt_ops["cx"], 16)
 
     def test_rustiq_upto_phase(self):
         """Check that Rustiq synthesis with ``upto_phase=True`` produces a correct