Repository: Qiskit/qiskit (Python)

Issue:
`PauliEvolutionGate` introduces sneaky complex part in parameter expression coefficients
### Environment

- **Qiskit version**:  1.3
- **Python version**: 
- **Operating system**: 


### What is happening?

An external repo that accesses an internal property of `ParameterExpression` started breaking with qiskit 1.3 for the official QOAO tutorial. This can be traced back to an unexpected complex coefficient in the `PauliEvolutionGate` parameter expression that builds the QAOA ansatz.

### How can we reproduce the issue?

The original issue was caught with this reproducing code (it accesses internal properties )

```
from qiskit.quantum_info import SparsePauliOp
from qiskit import QuantumCircuit
from qiskit.circuit.library import QAOAAnsatz
from qiskit.transpiler import PassManager
from qiskit.transpiler.passes import HighLevelSynthesis
from qiskit.circuit import ParameterExpression

cost_operator = SparsePauliOp.from_list([("IIZZ", -1), ("ZIIZ", 1), ("IZIZ", 2)])

cost_layer = QAOAAnsatz(
    cost_operator,
    mixer_operator=QuantumCircuit(4),
    initial_state=QuantumCircuit(4),
    reps=1,
).decompose()

pass_manager = PassManager([HighLevelSynthesis(basis_gates=["rzz"])])
circuit = pass_manager.run(cost_layer)

for inst in circuit.data:
    iop = inst.operation
    print("param expression", iop.params[0])
    print("symbol expression", iop.params[0]._symbol_expr)
```
The output with 1.2.4 is:

```
param expression -2.0*γ[0]
symbol expression -2.0*γ[0]
param expression 2.0*γ[0]
symbol expression 2.0*γ[0]
param expression 4.0*γ[0]
symbol expression 4.0*γ[0]
```

While 1.3.0+ shows:

```
param expression -2.0*γ[0]
symbol expression (-2.0 + 0.0*I)*γ[0]
param expression 2.0*γ[0]
symbol expression (2.0 + 0.0*I)*γ[0]
param expression 4.0*γ[0]
symbol expression (4.0 + 0.0*I)*γ[0]
```

But this issue can also be seen with a shorter snippet:

```
  time = Parameter("t")
  evo = PauliEvolutionGate(Z, time=time)

  angle = evo.definition.data[0].operation.params[0]
  print(angle.sympify())
```

### What should happen?

No complex coefficients should be found

### Any suggestions?

Working on a fix with @Cryoris


Discussion:

Patch:
diff --git a/qiskit/synthesis/evolution/product_formula.py b/qiskit/synthesis/evolution/product_formula.py
index b314d5ec5e62..980d8cdfefd9 100644
--- a/qiskit/synthesis/evolution/product_formula.py
+++ b/qiskit/synthesis/evolution/product_formula.py
@@ -191,12 +191,6 @@ def settings(self) -> dict[str, typing.Any]:
             "preserve_order": self.preserve_order,
         }
 
-    def _normalize_coefficients(
-        self, paulis: list[str | list[int], float | complex | ParameterExpression]
-    ) -> list[str | list[int] | ParameterValueType]:
-        """Ensure the coefficients are real (or parameter expressions)."""
-        return [[(op, qubits, real_or_fail(coeff)) for op, qubits, coeff in ops] for ops in paulis]
-
     def _custom_evolution(self, num_qubits, pauli_rotations):
         """Implement the evolution for the non-standard path.
 
diff --git a/qiskit/synthesis/evolution/suzuki_trotter.py b/qiskit/synthesis/evolution/suzuki_trotter.py
index 209f377351a7..8e2835f505b9 100644
--- a/qiskit/synthesis/evolution/suzuki_trotter.py
+++ b/qiskit/synthesis/evolution/suzuki_trotter.py
@@ -18,7 +18,9 @@
 import typing
 from collections.abc import Callable
 from itertools import chain
+import numpy as np
 
+from qiskit.circuit.parameterexpression import ParameterExpression
 from qiskit.circuit.quantumcircuit import QuantumCircuit
 from qiskit.quantum_info.operators import SparsePauliOp, Pauli
 from qiskit.utils.deprecation import deprecate_arg
@@ -157,7 +159,10 @@ def expand(
         time = evolution.time
 
         def to_sparse_list(operator):
-            paulis = (time * (2 / self.reps) * operator).to_sparse_list()
+            paulis = [
+                (pauli, indices, real_or_fail(coeff) * time * 2 / self.reps)
+                for pauli, indices, coeff in operator.to_sparse_list()
+            ]
             if not self.preserve_order:
                 return reorder_paulis(paulis)
 
@@ -171,9 +176,6 @@ def to_sparse_list(operator):
             # here would be the location to do so.
             non_commuting = [[op] for op in to_sparse_list(operators)]
 
-        # normalize coefficients, i.e. ensure they are float or ParameterExpression
-        non_commuting = self._normalize_coefficients(non_commuting)
-
         # we're already done here since Lie Trotter does not do any operator repetition
         product_formula = self._recurse(self.order, non_commuting)
         flattened = self.reps * list(chain.from_iterable(product_formula))
@@ -213,3 +215,18 @@ def _recurse(order, grouped_paulis):
                 ],
             )
             return outer + inner + outer
+
+
+def real_or_fail(value, tol=100):
+    """Return real if close, otherwise fail. Unbound parameters are left unchanged.
+
+    Based on NumPy's ``real_if_close``, i.e. ``tol`` is in terms of machine precision for float.
+    """
+    if isinstance(value, ParameterExpression):
+        return value
+
+    abstol = tol * np.finfo(float).eps
+    if abs(np.imag(value)) < abstol:
+        return np.real(value)
+
+    raise ValueError(f"Encountered complex value {value}, but expected real.")
diff --git a/releasenotes/notes/fix-pauli-sympify-ea9acceb2a923aff.yaml b/releasenotes/notes/fix-pauli-sympify-ea9acceb2a923aff.yaml
new file mode 100644
index 000000000000..248a09b6f88e
--- /dev/null
+++ b/releasenotes/notes/fix-pauli-sympify-ea9acceb2a923aff.yaml
@@ -0,0 +1,9 @@
+---
+fixes:
+  - |
+    Fixed an inconsistency in the circuit generated by a Pauli evolution synthesis
+    with :class:`.SuzukiTrotter` or :class:`.LieTrotter` (the default) method.
+    For parameterized evolution times, the resulting circuits contained parameters
+    with a spurious, zero complex part, which affected the output of
+    :meth:`.ParameterExpression.sympify`. The output now correctly is only real.
+    Fixed `#13642 <https://github.com/Qiskit/qiskit/pull/13642>`__.


Test patch:
diff --git a/test/python/circuit/library/test_evolution_gate.py b/test/python/circuit/library/test_evolution_gate.py
index 1b55ea920289..72acfc421a31 100644
--- a/test/python/circuit/library/test_evolution_gate.py
+++ b/test/python/circuit/library/test_evolution_gate.py
@@ -479,6 +479,20 @@ def atomic_evolution(pauli, time):
         decomposed = evo_gate.definition.decompose()
         self.assertEqual(decomposed.count_ops()["cx"], reps * 3 * 4)
 
+    def test_sympify_is_real(self):
+        """Test converting the parameters to sympy is real.
+
+        Regression test of #13642, where the parameters in the Pauli evolution had a spurious
+        zero complex part. Even though this is not noticable upon binding or printing the parameter,
+        it does affect the output of Parameter.sympify.
+        """
+        time = Parameter("t")
+        evo = PauliEvolutionGate(Z, time=time)
+
+        angle = evo.definition.data[0].operation.params[0]
+        expected = (2.0 * time).sympify()
+        self.assertEqual(expected, angle.sympify())
+
 
 def exact_atomic_evolution(circuit, pauli, time):
     """An exact atomic evolution for Suzuki-Trotter.