We are currently solving the following issue within our repository. Here is the issue text:
--- BEGIN ISSUE ---
SuzukiTrotter class should not accept odd orders
### Environment

- **Qiskit Terra version**: 
- **Python version**: 
- **Operating system**: 
It applies to all environments.

### What is happening?

The SuzukiTrotter class in `qiskit.synthesis.evolution` currently accepts any integer as input for the order. However the formula is not defined for odd orders (see definition below) and even though the code works on such input, the results do not make sense when the order is odd. 
![image](https://user-images.githubusercontent.com/5624856/142388673-acc5c6ce-1073-4103-9227-86d4fc79a63e.png)


### How can we reproduce the issue?

```python
from qiskit.opflow import X, Y
from qiskit.circuit.library import PauliEvolutionGate
from qiskit.synthesis import SuzukiTrotter
op = X + Y
time = 0.1
reps = 1
evo_gate = PauliEvolutionGate(op, time, synthesis=SuzukiTrotter(order=1, reps=reps))
```

### What should happen?

It should raise an error message and do not accept odd integers as input for the order.

### Any suggestions?

_No response_

--- END ISSUE ---

Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
--- BEGIN FILES ---
Path: `qiskit/synthesis/evolution/suzuki_trotter.py`
Content:
```
1 # This code is part of Qiskit.
2 #
3 # (C) Copyright IBM 2021.
4 #
5 # This code is licensed under the Apache License, Version 2.0. You may
6 # obtain a copy of this license in the LICENSE.txt file in the root directory
7 # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
8 #
9 # Any modifications or derivative works of this code must retain this
10 # copyright notice, and modified files need to carry a notice indicating
11 # that they have been altered from the originals.
12 
13 """The Suzuki-Trotter product formula."""
14 
15 from typing import Callable, Optional, Union
16 import numpy as np
17 from qiskit.circuit.quantumcircuit import QuantumCircuit
18 from qiskit.quantum_info.operators import SparsePauliOp, Pauli
19 
20 from .product_formula import ProductFormula
21 
22 
23 class SuzukiTrotter(ProductFormula):
24     r"""The (higher order) Suzuki-Trotter product formula.
25 
26     The Suzuki-Trotter formulas improve the error of the Lie-Trotter approximation.
27     For example, the second order decomposition is
28 
29     .. math::
30 
31         e^{A + B} \approx e^{B/2} e^{A} e^{B/2}.
32 
33     Higher order decompositions are based on recursions, see Ref. [1] for more details.
34 
35     In this implementation, the operators are provided as sum terms of a Pauli operator.
36     For example, in the second order Suzuki-Trotter decomposition we approximate
37 
38     .. math::
39 
40         e^{-it(XX + ZZ)} = e^{-it/2 ZZ}e^{-it XX}e^{-it/2 ZZ} + \mathcal{O}(t^2).
41 
42     References:
43         [1]: D. Berry, G. Ahokas, R. Cleve and B. Sanders,
44         "Efficient quantum algorithms for simulating sparse Hamiltonians" (2006).
45         `arXiv:quant-ph/0508139 <https://arxiv.org/abs/quant-ph/0508139>`_
46     """
47 
48     def __init__(
49         self,
50         order: int = 2,
51         reps: int = 1,
52         insert_barriers: bool = False,
53         cx_structure: str = "chain",
54         atomic_evolution: Optional[
55             Callable[[Union[Pauli, SparsePauliOp], float], QuantumCircuit]
56         ] = None,
57     ) -> None:
58         """
59         Args:
60             order: The order of the product formula.
61             reps: The number of time steps.
62             insert_barriers: Whether to insert barriers between the atomic evolutions.
63             cx_structure: How to arrange the CX gates for the Pauli evolutions, can be "chain",
64                 where next neighbor connections are used, or "fountain", where all qubits are connected
65                 to one.
66             atomic_evolution: A function to construct the circuit for the evolution of single
67                 Pauli string. Per default, a single Pauli evolution is decomopsed in a CX chain
68                 and a single qubit Z rotation.
69         """
70         super().__init__(order, reps, insert_barriers, cx_structure, atomic_evolution)
71 
72     def synthesize(self, evolution):
73         # get operators and time to evolve
74         operators = evolution.operator
75         time = evolution.time
76 
77         if not isinstance(operators, list):
78             pauli_list = [(Pauli(op), np.real(coeff)) for op, coeff in operators.to_list()]
79         else:
80             pauli_list = [(op, 1) for op in operators]
81 
82         ops_to_evolve = self._recurse(self.order, time / self.reps, pauli_list)
83 
84         # construct the evolution circuit
85         single_rep = QuantumCircuit(operators[0].num_qubits)
86         first_barrier = False
87 
88         for op, coeff in ops_to_evolve:
89             # add barriers
90             if first_barrier:
91                 if self.insert_barriers:
92                     single_rep.barrier()
93             else:
94                 first_barrier = True
95 
96             single_rep.compose(self.atomic_evolution(op, coeff), wrap=True, inplace=True)
97 
98         evolution_circuit = QuantumCircuit(operators[0].num_qubits)
99         first_barrier = False
100 
101         for _ in range(self.reps):
102             # add barriers
103             if first_barrier:
104                 if self.insert_barriers:
105                     single_rep.barrier()
106             else:
107                 first_barrier = True
108 
109             evolution_circuit.compose(single_rep, inplace=True)
110 
111         return evolution_circuit
112 
113     @staticmethod
114     def _recurse(order, time, pauli_list):
115         if order == 1:
116             return pauli_list
117 
118         elif order == 2:
119             halves = [(op, coeff * time / 2) for op, coeff in pauli_list[:-1]]
120             full = [(pauli_list[-1][0], time * pauli_list[-1][1])]
121             return halves + full + list(reversed(halves))
122 
123         else:
124             reduction = 1 / (4 - 4 ** (1 / (order - 1)))
125             outer = 2 * SuzukiTrotter._recurse(
126                 order - 2, time=reduction * time, pauli_list=pauli_list
127             )
128             inner = SuzukiTrotter._recurse(
129                 order - 2, time=(1 - 4 * reduction) * time, pauli_list=pauli_list
130             )
131             return outer + inner + outer
132
```
--- END FILES ---

Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.

Here is an example:

```diff
diff --git a/examples/server_async.py b/examples/server_async.py
--- a/examples/server_async.py
+++ b/examples/server_async.py
@@ -313,4 +313,4 @@
 
 
 if __name__ == "__main__":
-    asyncio.run(run_async_server("."), debug=True)
+    asyncio.run(run_async_server(), debug=True)
diff --git a/examples/server_sync.py b/examples/server_sync.py
--- a/examples/server_sync.py
+++ b/examples/server_sync.py
@@ -313,5 +313,5 @@
 
 
 if __name__ == "__main__":
-    server = run_sync_server(".")
+    server = run_sync_server()
     server.shutdown()

```

diff --git a/qiskit/synthesis/evolution/suzuki_trotter.py b/qiskit/synthesis/evolution/suzuki_trotter.py
--- a/qiskit/synthesis/evolution/suzuki_trotter.py
+++ b/qiskit/synthesis/evolution/suzuki_trotter.py
@@ -13,7 +13,10 @@
 """The Suzuki-Trotter product formula."""
 
 from typing import Callable, Optional, Union
+
+import warnings
 import numpy as np
+
 from qiskit.circuit.quantumcircuit import QuantumCircuit
 from qiskit.quantum_info.operators import SparsePauliOp, Pauli
 
@@ -61,12 +64,24 @@
             reps: The number of time steps.
             insert_barriers: Whether to insert barriers between the atomic evolutions.
             cx_structure: How to arrange the CX gates for the Pauli evolutions, can be "chain",
-                where next neighbor connections are used, or "fountain", where all qubits are connected
-                to one.
+                where next neighbor connections are used, or "fountain", where all qubits are
+                connected to one.
             atomic_evolution: A function to construct the circuit for the evolution of single
                 Pauli string. Per default, a single Pauli evolution is decomopsed in a CX chain
                 and a single qubit Z rotation.
         """
+        if order % 2 == 1:
+            warnings.warn(
+                "SuzukiTrotter for odd orders is deprecated as of 0.20.0, and will be "
+                "removed no earlier than 3 months after that release date. Suzuki "
+                "product formulae are symmetric and therefore only defined for even"
+                "orders.",
+                DeprecationWarning,
+                stacklevel=2,
+            )
+            # TODO replace deprecation warning by the following error and add unit test for odd
+            # raise ValueError("Suzuki product formulae are symmetric and therefore only defined "
+            #                  "for even orders.")
         super().__init__(order, reps, insert_barriers, cx_structure, atomic_evolution)
 
     def synthesize(self, evolution):

{"golden_diff": "diff --git a/qiskit/synthesis/evolution/suzuki_trotter.py b/qiskit/synthesis/evolution/suzuki_trotter.py\n--- a/qiskit/synthesis/evolution/suzuki_trotter.py\n+++ b/qiskit/synthesis/evolution/suzuki_trotter.py\n@@ -13,7 +13,10 @@\n \"\"\"The Suzuki-Trotter product formula.\"\"\"\n \n from typing import Callable, Optional, Union\n+\n+import warnings\n import numpy as np\n+\n from qiskit.circuit.quantumcircuit import QuantumCircuit\n from qiskit.quantum_info.operators import SparsePauliOp, Pauli\n \n@@ -61,12 +64,24 @@\n             reps: The number of time steps.\n             insert_barriers: Whether to insert barriers between the atomic evolutions.\n             cx_structure: How to arrange the CX gates for the Pauli evolutions, can be \"chain\",\n-                where next neighbor connections are used, or \"fountain\", where all qubits are connected\n-                to one.\n+                where next neighbor connections are used, or \"fountain\", where all qubits are\n+                connected to one.\n             atomic_evolution: A function to construct the circuit for the evolution of single\n                 Pauli string. Per default, a single Pauli evolution is decomopsed in a CX chain\n                 and a single qubit Z rotation.\n         \"\"\"\n+        if order % 2 == 1:\n+            warnings.warn(\n+                \"SuzukiTrotter for odd orders is deprecated as of 0.20.0, and will be \"\n+                \"removed no earlier than 3 months after that release date. Suzuki \"\n+                \"product formulae are symmetric and therefore only defined for even\"\n+                \"orders.\",\n+                DeprecationWarning,\n+                stacklevel=2,\n+            )\n+            # TODO replace deprecation warning by the following error and add unit test for odd\n+            # raise ValueError(\"Suzuki product formulae are symmetric and therefore only defined \"\n+            #                  \"for even orders.\")\n         super().__init__(order, reps, insert_barriers, cx_structure, atomic_evolution)\n \n     def synthesize(self, evolution):\n", "issue": "SuzukiTrotter class should not accept odd orders\n### Environment\n\n- **Qiskit Terra version**: \r\n- **Python version**: \r\n- **Operating system**: \r\nIt applies to all environments.\n\n### What is happening?\n\nThe SuzukiTrotter class in `qiskit.synthesis.evolution` currently accepts any integer as input for the order. However the formula is not defined for odd orders (see definition below) and even though the code works on such input, the results do not make sense when the order is odd. \r\n![image](https://user-images.githubusercontent.com/5624856/142388673-acc5c6ce-1073-4103-9227-86d4fc79a63e.png)\r\n\n\n### How can we reproduce the issue?\n\n```python\r\nfrom qiskit.opflow import X, Y\r\nfrom qiskit.circuit.library import PauliEvolutionGate\r\nfrom qiskit.synthesis import SuzukiTrotter\r\nop = X + Y\r\ntime = 0.1\r\nreps = 1\r\nevo_gate = PauliEvolutionGate(op, time, synthesis=SuzukiTrotter(order=1, reps=reps))\r\n```\n\n### What should happen?\n\nIt should raise an error message and do not accept odd integers as input for the order.\n\n### Any suggestions?\n\n_No response_\n", "before_files": [{"content": "# This code is part of Qiskit.\n#\n# (C) Copyright IBM 2021.\n#\n# This code is licensed under the Apache License, Version 2.0. You may\n# obtain a copy of this license in the LICENSE.txt file in the root directory\n# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.\n#\n# Any modifications or derivative works of this code must retain this\n# copyright notice, and modified files need to carry a notice indicating\n# that they have been altered from the originals.\n\n\"\"\"The Suzuki-Trotter product formula.\"\"\"\n\nfrom typing import Callable, Optional, Union\nimport numpy as np\nfrom qiskit.circuit.quantumcircuit import QuantumCircuit\nfrom qiskit.quantum_info.operators import SparsePauliOp, Pauli\n\nfrom .product_formula import ProductFormula\n\n\nclass SuzukiTrotter(ProductFormula):\n    r\"\"\"The (higher order) Suzuki-Trotter product formula.\n\n    The Suzuki-Trotter formulas improve the error of the Lie-Trotter approximation.\n    For example, the second order decomposition is\n\n    .. math::\n\n        e^{A + B} \\approx e^{B/2} e^{A} e^{B/2}.\n\n    Higher order decompositions are based on recursions, see Ref. [1] for more details.\n\n    In this implementation, the operators are provided as sum terms of a Pauli operator.\n    For example, in the second order Suzuki-Trotter decomposition we approximate\n\n    .. math::\n\n        e^{-it(XX + ZZ)} = e^{-it/2 ZZ}e^{-it XX}e^{-it/2 ZZ} + \\mathcal{O}(t^2).\n\n    References:\n        [1]: D. Berry, G. Ahokas, R. Cleve and B. Sanders,\n        \"Efficient quantum algorithms for simulating sparse Hamiltonians\" (2006).\n        `arXiv:quant-ph/0508139 <https://arxiv.org/abs/quant-ph/0508139>`_\n    \"\"\"\n\n    def __init__(\n        self,\n        order: int = 2,\n        reps: int = 1,\n        insert_barriers: bool = False,\n        cx_structure: str = \"chain\",\n        atomic_evolution: Optional[\n            Callable[[Union[Pauli, SparsePauliOp], float], QuantumCircuit]\n        ] = None,\n    ) -> None:\n        \"\"\"\n        Args:\n            order: The order of the product formula.\n            reps: The number of time steps.\n            insert_barriers: Whether to insert barriers between the atomic evolutions.\n            cx_structure: How to arrange the CX gates for the Pauli evolutions, can be \"chain\",\n                where next neighbor connections are used, or \"fountain\", where all qubits are connected\n                to one.\n            atomic_evolution: A function to construct the circuit for the evolution of single\n                Pauli string. Per default, a single Pauli evolution is decomopsed in a CX chain\n                and a single qubit Z rotation.\n        \"\"\"\n        super().__init__(order, reps, insert_barriers, cx_structure, atomic_evolution)\n\n    def synthesize(self, evolution):\n        # get operators and time to evolve\n        operators = evolution.operator\n        time = evolution.time\n\n        if not isinstance(operators, list):\n            pauli_list = [(Pauli(op), np.real(coeff)) for op, coeff in operators.to_list()]\n        else:\n            pauli_list = [(op, 1) for op in operators]\n\n        ops_to_evolve = self._recurse(self.order, time / self.reps, pauli_list)\n\n        # construct the evolution circuit\n        single_rep = QuantumCircuit(operators[0].num_qubits)\n        first_barrier = False\n\n        for op, coeff in ops_to_evolve:\n            # add barriers\n            if first_barrier:\n                if self.insert_barriers:\n                    single_rep.barrier()\n            else:\n                first_barrier = True\n\n            single_rep.compose(self.atomic_evolution(op, coeff), wrap=True, inplace=True)\n\n        evolution_circuit = QuantumCircuit(operators[0].num_qubits)\n        first_barrier = False\n\n        for _ in range(self.reps):\n            # add barriers\n            if first_barrier:\n                if self.insert_barriers:\n                    single_rep.barrier()\n            else:\n                first_barrier = True\n\n            evolution_circuit.compose(single_rep, inplace=True)\n\n        return evolution_circuit\n\n    @staticmethod\n    def _recurse(order, time, pauli_list):\n        if order == 1:\n            return pauli_list\n\n        elif order == 2:\n            halves = [(op, coeff * time / 2) for op, coeff in pauli_list[:-1]]\n            full = [(pauli_list[-1][0], time * pauli_list[-1][1])]\n            return halves + full + list(reversed(halves))\n\n        else:\n            reduction = 1 / (4 - 4 ** (1 / (order - 1)))\n            outer = 2 * SuzukiTrotter._recurse(\n                order - 2, time=reduction * time, pauli_list=pauli_list\n            )\n            inner = SuzukiTrotter._recurse(\n                order - 2, time=(1 - 4 * reduction) * time, pauli_list=pauli_list\n            )\n            return outer + inner + outer\n", "path": "qiskit/synthesis/evolution/suzuki_trotter.py"}], "after_files": [{"content": "# This code is part of Qiskit.\n#\n# (C) Copyright IBM 2021.\n#\n# This code is licensed under the Apache License, Version 2.0. You may\n# obtain a copy of this license in the LICENSE.txt file in the root directory\n# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.\n#\n# Any modifications or derivative works of this code must retain this\n# copyright notice, and modified files need to carry a notice indicating\n# that they have been altered from the originals.\n\n\"\"\"The Suzuki-Trotter product formula.\"\"\"\n\nfrom typing import Callable, Optional, Union\n\nimport warnings\nimport numpy as np\n\nfrom qiskit.circuit.quantumcircuit import QuantumCircuit\nfrom qiskit.quantum_info.operators import SparsePauliOp, Pauli\n\nfrom .product_formula import ProductFormula\n\n\nclass SuzukiTrotter(ProductFormula):\n    r\"\"\"The (higher order) Suzuki-Trotter product formula.\n\n    The Suzuki-Trotter formulas improve the error of the Lie-Trotter approximation.\n    For example, the second order decomposition is\n\n    .. math::\n\n        e^{A + B} \\approx e^{B/2} e^{A} e^{B/2}.\n\n    Higher order decompositions are based on recursions, see Ref. [1] for more details.\n\n    In this implementation, the operators are provided as sum terms of a Pauli operator.\n    For example, in the second order Suzuki-Trotter decomposition we approximate\n\n    .. math::\n\n        e^{-it(XX + ZZ)} = e^{-it/2 ZZ}e^{-it XX}e^{-it/2 ZZ} + \\mathcal{O}(t^2).\n\n    References:\n        [1]: D. Berry, G. Ahokas, R. Cleve and B. Sanders,\n        \"Efficient quantum algorithms for simulating sparse Hamiltonians\" (2006).\n        `arXiv:quant-ph/0508139 <https://arxiv.org/abs/quant-ph/0508139>`_\n    \"\"\"\n\n    def __init__(\n        self,\n        order: int = 2,\n        reps: int = 1,\n        insert_barriers: bool = False,\n        cx_structure: str = \"chain\",\n        atomic_evolution: Optional[\n            Callable[[Union[Pauli, SparsePauliOp], float], QuantumCircuit]\n        ] = None,\n    ) -> None:\n        \"\"\"\n        Args:\n            order: The order of the product formula.\n            reps: The number of time steps.\n            insert_barriers: Whether to insert barriers between the atomic evolutions.\n            cx_structure: How to arrange the CX gates for the Pauli evolutions, can be \"chain\",\n                where next neighbor connections are used, or \"fountain\", where all qubits are\n                connected to one.\n            atomic_evolution: A function to construct the circuit for the evolution of single\n                Pauli string. Per default, a single Pauli evolution is decomopsed in a CX chain\n                and a single qubit Z rotation.\n        \"\"\"\n        if order % 2 == 1:\n            warnings.warn(\n                \"SuzukiTrotter for odd orders is deprecated as of 0.20.0, and will be \"\n                \"removed no earlier than 3 months after that release date. Suzuki \"\n                \"product formulae are symmetric and therefore only defined for even\"\n                \"orders.\",\n                DeprecationWarning,\n                stacklevel=2,\n            )\n            # TODO replace deprecation warning by the following error and add unit test for odd\n            # raise ValueError(\"Suzuki product formulae are symmetric and therefore only defined \"\n            #                  \"for even orders.\")\n        super().__init__(order, reps, insert_barriers, cx_structure, atomic_evolution)\n\n    def synthesize(self, evolution):\n        # get operators and time to evolve\n        operators = evolution.operator\n        time = evolution.time\n\n        if not isinstance(operators, list):\n            pauli_list = [(Pauli(op), np.real(coeff)) for op, coeff in operators.to_list()]\n        else:\n            pauli_list = [(op, 1) for op in operators]\n\n        ops_to_evolve = self._recurse(self.order, time / self.reps, pauli_list)\n\n        # construct the evolution circuit\n        single_rep = QuantumCircuit(operators[0].num_qubits)\n        first_barrier = False\n\n        for op, coeff in ops_to_evolve:\n            # add barriers\n            if first_barrier:\n                if self.insert_barriers:\n                    single_rep.barrier()\n            else:\n                first_barrier = True\n\n            single_rep.compose(self.atomic_evolution(op, coeff), wrap=True, inplace=True)\n\n        evolution_circuit = QuantumCircuit(operators[0].num_qubits)\n        first_barrier = False\n\n        for _ in range(self.reps):\n            # add barriers\n            if first_barrier:\n                if self.insert_barriers:\n                    single_rep.barrier()\n            else:\n                first_barrier = True\n\n            evolution_circuit.compose(single_rep, inplace=True)\n\n        return evolution_circuit\n\n    @staticmethod\n    def _recurse(order, time, pauli_list):\n        if order == 1:\n            return pauli_list\n\n        elif order == 2:\n            halves = [(op, coeff * time / 2) for op, coeff in pauli_list[:-1]]\n            full = [(pauli_list[-1][0], time * pauli_list[-1][1])]\n            return halves + full + list(reversed(halves))\n\n        else:\n            reduction = 1 / (4 - 4 ** (1 / (order - 1)))\n            outer = 2 * SuzukiTrotter._recurse(\n                order - 2, time=reduction * time, pauli_list=pauli_list\n            )\n            inner = SuzukiTrotter._recurse(\n                order - 2, time=(1 - 4 * reduction) * time, pauli_list=pauli_list\n            )\n            return outer + inner + outer\n", "path": "qiskit/synthesis/evolution/suzuki_trotter.py"}]}