We are currently solving the following issue within our repository. Here is the issue text:
--- BEGIN ISSUE ---
Pauli operator deserialization, and commutation checking
When you serialize and then deserialize a Pauli operator, you get a different instance, which might not be too surprising
```python
cirq.read_json(json_text=cirq.to_json(cirq.Z)) is cirq.Z

False
```
This shouldn't be a problem, however Paulis are checking commutation using `is` instead of `==`, so commutation with deserialized Paulis is broken.

--- END ISSUE ---

Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
--- BEGIN FILES ---
Path: `cirq/ops/pauli_gates.py`
Content:
```
1 # Copyright 2018 The Cirq Developers
2 #
3 # Licensed under the Apache License, Version 2.0 (the "License");
4 # you may not use this file except in compliance with the License.
5 # You may obtain a copy of the License at
6 #
7 #     https://www.apache.org/licenses/LICENSE-2.0
8 #
9 # Unless required by applicable law or agreed to in writing, software
10 # distributed under the License is distributed on an "AS IS" BASIS,
11 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 # See the License for the specific language governing permissions and
13 # limitations under the License.
14 import abc
15 from typing import Any, cast, Tuple, TYPE_CHECKING, Union
16 
17 from cirq import value
18 from cirq._doc import document
19 from cirq.ops import common_gates, raw_types, identity
20 from cirq.type_workarounds import NotImplementedType
21 
22 
23 if TYPE_CHECKING:
24     import cirq
25     from cirq.ops.pauli_string import SingleQubitPauliStringGateOperation
26 
27 
28 class Pauli(raw_types.Gate, metaclass=abc.ABCMeta):
29     """Represents the Pauli gates.
30 
31     This is an abstract class with no public subclasses. The only instances
32     of private subclasses are the X, Y, or Z Pauli gates defined below.
33     """
34     _XYZ = None  # type: Tuple[Pauli, Pauli, Pauli]
35 
36     @staticmethod
37     def by_index(index: int) -> 'Pauli':
38         return Pauli._XYZ[index % 3]
39 
40     @staticmethod
41     def by_relative_index(p: 'Pauli', relative_index: int) -> 'Pauli':
42         return Pauli._XYZ[(p._index + relative_index) % 3]
43 
44     def __init__(self, index: int, name: str) -> None:
45         self._index = index
46         self._name = name
47 
48     def num_qubits(self):
49         return 1
50 
51     def _commutes_(self, other: Any,
52                    atol: float) -> Union[bool, NotImplementedType, None]:
53         if not isinstance(other, Pauli):
54             return NotImplemented
55         return self is other
56 
57     def third(self, second: 'Pauli') -> 'Pauli':
58         return Pauli._XYZ[(-self._index - second._index) % 3]
59 
60     def relative_index(self, second: 'Pauli') -> int:
61         """Relative index of self w.r.t. second in the (X, Y, Z) cycle."""
62         return (self._index - second._index + 1) % 3 - 1
63 
64     def phased_pauli_product(
65             self, other: Union['cirq.Pauli', 'identity.IdentityGate']
66     ) -> Tuple[complex, Union['cirq.Pauli', 'identity.IdentityGate']]:
67         if self == other:
68             return 1, identity.I
69         if other is identity.I:
70             return 1, self
71         return 1j**cast(Pauli, other).relative_index(self), self.third(
72             cast(Pauli, other))
73 
74     def __gt__(self, other):
75         if not isinstance(other, Pauli):
76             return NotImplemented
77         return (self._index - other._index) % 3 == 1
78 
79     def __lt__(self, other):
80         if not isinstance(other, Pauli):
81             return NotImplemented
82         return (other._index - self._index) % 3 == 1
83 
84     def on(self, *qubits: 'cirq.Qid') -> 'SingleQubitPauliStringGateOperation':
85         """Returns an application of this gate to the given qubits.
86 
87         Args:
88             *qubits: The collection of qubits to potentially apply the gate to.
89         """
90         if len(qubits) != 1:
91             raise ValueError(
92                 'Expected a single qubit, got <{!r}>.'.format(qubits))
93         from cirq.ops.pauli_string import SingleQubitPauliStringGateOperation
94         return SingleQubitPauliStringGateOperation(self, qubits[0])
95 
96     @property
97     def _canonical_exponent(self):
98         """Overrides EigenGate._canonical_exponent in subclasses."""
99         return 1
100 
101 
102 class _PauliX(Pauli, common_gates.XPowGate):
103 
104     def __init__(self):
105         Pauli.__init__(self, index=0, name='X')
106         common_gates.XPowGate.__init__(self, exponent=1.0)
107 
108     def __pow__(self: '_PauliX',
109                 exponent: value.TParamVal) -> common_gates.XPowGate:
110         return common_gates.XPowGate(exponent=exponent)
111 
112     def _with_exponent(self: '_PauliX',
113                        exponent: value.TParamVal) -> common_gates.XPowGate:
114         return self.__pow__(exponent)
115 
116     @classmethod
117     def _from_json_dict_(cls, exponent, global_shift, **kwargs):
118         assert global_shift == 0
119         assert exponent == 1
120         return cls()
121 
122 
123 class _PauliY(Pauli, common_gates.YPowGate):
124 
125     def __init__(self):
126         Pauli.__init__(self, index=1, name='Y')
127         common_gates.YPowGate.__init__(self, exponent=1.0)
128 
129     def __pow__(self: '_PauliY',
130                 exponent: value.TParamVal) -> common_gates.YPowGate:
131         return common_gates.YPowGate(exponent=exponent)
132 
133     def _with_exponent(self: '_PauliY',
134                        exponent: value.TParamVal) -> common_gates.YPowGate:
135         return self.__pow__(exponent)
136 
137     @classmethod
138     def _from_json_dict_(cls, exponent, global_shift, **kwargs):
139         assert global_shift == 0
140         assert exponent == 1
141         return cls()
142 
143 
144 class _PauliZ(Pauli, common_gates.ZPowGate):
145 
146     def __init__(self):
147         Pauli.__init__(self, index=2, name='Z')
148         common_gates.ZPowGate.__init__(self, exponent=1.0)
149 
150     def __pow__(self: '_PauliZ',
151                 exponent: value.TParamVal) -> common_gates.ZPowGate:
152         return common_gates.ZPowGate(exponent=exponent)
153 
154     def _with_exponent(self: '_PauliZ',
155                        exponent: value.TParamVal) -> common_gates.ZPowGate:
156         return self.__pow__(exponent)
157 
158     @classmethod
159     def _from_json_dict_(cls, exponent, global_shift, **kwargs):
160         assert global_shift == 0
161         assert exponent == 1
162         return cls()
163 
164 
165 X = _PauliX()
166 document(
167     X, """The Pauli X gate.
168 
169     Matrix:
170 
171         [[0, 1],
172          [1, 0]]
173     """)
174 
175 Y = _PauliY()
176 document(
177     Y, """The Pauli Y gate.
178 
179     Matrix:
180 
181         [[0, -i],
182          [i, 0]]
183     """)
184 
185 Z = _PauliZ()
186 document(
187     Z, """The Pauli Z gate.
188 
189     Matrix:
190 
191         [[1, 0],
192          [0, -1]]
193     """)
194 
195 Pauli._XYZ = (X, Y, Z)
196
```
--- 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/cirq/ops/pauli_gates.py b/cirq/ops/pauli_gates.py
--- a/cirq/ops/pauli_gates.py
+++ b/cirq/ops/pauli_gates.py
@@ -117,7 +117,7 @@
     def _from_json_dict_(cls, exponent, global_shift, **kwargs):
         assert global_shift == 0
         assert exponent == 1
-        return cls()
+        return Pauli._XYZ[0]
 
 
 class _PauliY(Pauli, common_gates.YPowGate):
@@ -138,7 +138,7 @@
     def _from_json_dict_(cls, exponent, global_shift, **kwargs):
         assert global_shift == 0
         assert exponent == 1
-        return cls()
+        return Pauli._XYZ[1]
 
 
 class _PauliZ(Pauli, common_gates.ZPowGate):
@@ -159,7 +159,7 @@
     def _from_json_dict_(cls, exponent, global_shift, **kwargs):
         assert global_shift == 0
         assert exponent == 1
-        return cls()
+        return Pauli._XYZ[2]
 
 
 X = _PauliX()

{"golden_diff": "diff --git a/cirq/ops/pauli_gates.py b/cirq/ops/pauli_gates.py\n--- a/cirq/ops/pauli_gates.py\n+++ b/cirq/ops/pauli_gates.py\n@@ -117,7 +117,7 @@\n     def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n         assert global_shift == 0\n         assert exponent == 1\n-        return cls()\n+        return Pauli._XYZ[0]\n \n \n class _PauliY(Pauli, common_gates.YPowGate):\n@@ -138,7 +138,7 @@\n     def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n         assert global_shift == 0\n         assert exponent == 1\n-        return cls()\n+        return Pauli._XYZ[1]\n \n \n class _PauliZ(Pauli, common_gates.ZPowGate):\n@@ -159,7 +159,7 @@\n     def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n         assert global_shift == 0\n         assert exponent == 1\n-        return cls()\n+        return Pauli._XYZ[2]\n \n \n X = _PauliX()\n", "issue": "Pauli operator deserialization, and commutation checking\nWhen you serialize and then deserialize a Pauli operator, you get a different instance, which might not be too surprising\r\n```python\r\ncirq.read_json(json_text=cirq.to_json(cirq.Z)) is cirq.Z\r\n\r\nFalse\r\n```\r\nThis shouldn't be a problem, however Paulis are checking commutation using `is` instead of `==`, so commutation with deserialized Paulis is broken.\n", "before_files": [{"content": "# Copyright 2018 The Cirq Developers\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\nimport abc\nfrom typing import Any, cast, Tuple, TYPE_CHECKING, Union\n\nfrom cirq import value\nfrom cirq._doc import document\nfrom cirq.ops import common_gates, raw_types, identity\nfrom cirq.type_workarounds import NotImplementedType\n\n\nif TYPE_CHECKING:\n    import cirq\n    from cirq.ops.pauli_string import SingleQubitPauliStringGateOperation\n\n\nclass Pauli(raw_types.Gate, metaclass=abc.ABCMeta):\n    \"\"\"Represents the Pauli gates.\n\n    This is an abstract class with no public subclasses. The only instances\n    of private subclasses are the X, Y, or Z Pauli gates defined below.\n    \"\"\"\n    _XYZ = None  # type: Tuple[Pauli, Pauli, Pauli]\n\n    @staticmethod\n    def by_index(index: int) -> 'Pauli':\n        return Pauli._XYZ[index % 3]\n\n    @staticmethod\n    def by_relative_index(p: 'Pauli', relative_index: int) -> 'Pauli':\n        return Pauli._XYZ[(p._index + relative_index) % 3]\n\n    def __init__(self, index: int, name: str) -> None:\n        self._index = index\n        self._name = name\n\n    def num_qubits(self):\n        return 1\n\n    def _commutes_(self, other: Any,\n                   atol: float) -> Union[bool, NotImplementedType, None]:\n        if not isinstance(other, Pauli):\n            return NotImplemented\n        return self is other\n\n    def third(self, second: 'Pauli') -> 'Pauli':\n        return Pauli._XYZ[(-self._index - second._index) % 3]\n\n    def relative_index(self, second: 'Pauli') -> int:\n        \"\"\"Relative index of self w.r.t. second in the (X, Y, Z) cycle.\"\"\"\n        return (self._index - second._index + 1) % 3 - 1\n\n    def phased_pauli_product(\n            self, other: Union['cirq.Pauli', 'identity.IdentityGate']\n    ) -> Tuple[complex, Union['cirq.Pauli', 'identity.IdentityGate']]:\n        if self == other:\n            return 1, identity.I\n        if other is identity.I:\n            return 1, self\n        return 1j**cast(Pauli, other).relative_index(self), self.third(\n            cast(Pauli, other))\n\n    def __gt__(self, other):\n        if not isinstance(other, Pauli):\n            return NotImplemented\n        return (self._index - other._index) % 3 == 1\n\n    def __lt__(self, other):\n        if not isinstance(other, Pauli):\n            return NotImplemented\n        return (other._index - self._index) % 3 == 1\n\n    def on(self, *qubits: 'cirq.Qid') -> 'SingleQubitPauliStringGateOperation':\n        \"\"\"Returns an application of this gate to the given qubits.\n\n        Args:\n            *qubits: The collection of qubits to potentially apply the gate to.\n        \"\"\"\n        if len(qubits) != 1:\n            raise ValueError(\n                'Expected a single qubit, got <{!r}>.'.format(qubits))\n        from cirq.ops.pauli_string import SingleQubitPauliStringGateOperation\n        return SingleQubitPauliStringGateOperation(self, qubits[0])\n\n    @property\n    def _canonical_exponent(self):\n        \"\"\"Overrides EigenGate._canonical_exponent in subclasses.\"\"\"\n        return 1\n\n\nclass _PauliX(Pauli, common_gates.XPowGate):\n\n    def __init__(self):\n        Pauli.__init__(self, index=0, name='X')\n        common_gates.XPowGate.__init__(self, exponent=1.0)\n\n    def __pow__(self: '_PauliX',\n                exponent: value.TParamVal) -> common_gates.XPowGate:\n        return common_gates.XPowGate(exponent=exponent)\n\n    def _with_exponent(self: '_PauliX',\n                       exponent: value.TParamVal) -> common_gates.XPowGate:\n        return self.__pow__(exponent)\n\n    @classmethod\n    def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n        assert global_shift == 0\n        assert exponent == 1\n        return cls()\n\n\nclass _PauliY(Pauli, common_gates.YPowGate):\n\n    def __init__(self):\n        Pauli.__init__(self, index=1, name='Y')\n        common_gates.YPowGate.__init__(self, exponent=1.0)\n\n    def __pow__(self: '_PauliY',\n                exponent: value.TParamVal) -> common_gates.YPowGate:\n        return common_gates.YPowGate(exponent=exponent)\n\n    def _with_exponent(self: '_PauliY',\n                       exponent: value.TParamVal) -> common_gates.YPowGate:\n        return self.__pow__(exponent)\n\n    @classmethod\n    def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n        assert global_shift == 0\n        assert exponent == 1\n        return cls()\n\n\nclass _PauliZ(Pauli, common_gates.ZPowGate):\n\n    def __init__(self):\n        Pauli.__init__(self, index=2, name='Z')\n        common_gates.ZPowGate.__init__(self, exponent=1.0)\n\n    def __pow__(self: '_PauliZ',\n                exponent: value.TParamVal) -> common_gates.ZPowGate:\n        return common_gates.ZPowGate(exponent=exponent)\n\n    def _with_exponent(self: '_PauliZ',\n                       exponent: value.TParamVal) -> common_gates.ZPowGate:\n        return self.__pow__(exponent)\n\n    @classmethod\n    def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n        assert global_shift == 0\n        assert exponent == 1\n        return cls()\n\n\nX = _PauliX()\ndocument(\n    X, \"\"\"The Pauli X gate.\n\n    Matrix:\n\n        [[0, 1],\n         [1, 0]]\n    \"\"\")\n\nY = _PauliY()\ndocument(\n    Y, \"\"\"The Pauli Y gate.\n\n    Matrix:\n\n        [[0, -i],\n         [i, 0]]\n    \"\"\")\n\nZ = _PauliZ()\ndocument(\n    Z, \"\"\"The Pauli Z gate.\n\n    Matrix:\n\n        [[1, 0],\n         [0, -1]]\n    \"\"\")\n\nPauli._XYZ = (X, Y, Z)\n", "path": "cirq/ops/pauli_gates.py"}], "after_files": [{"content": "# Copyright 2018 The Cirq Developers\n#\n# Licensed under the Apache License, Version 2.0 (the \"License\");\n# you may not use this file except in compliance with the License.\n# You may obtain a copy of the License at\n#\n#     https://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing, software\n# distributed under the License is distributed on an \"AS IS\" BASIS,\n# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n# See the License for the specific language governing permissions and\n# limitations under the License.\nimport abc\nfrom typing import Any, cast, Tuple, TYPE_CHECKING, Union\n\nfrom cirq import value\nfrom cirq._doc import document\nfrom cirq.ops import common_gates, raw_types, identity\nfrom cirq.type_workarounds import NotImplementedType\n\n\nif TYPE_CHECKING:\n    import cirq\n    from cirq.ops.pauli_string import SingleQubitPauliStringGateOperation\n\n\nclass Pauli(raw_types.Gate, metaclass=abc.ABCMeta):\n    \"\"\"Represents the Pauli gates.\n\n    This is an abstract class with no public subclasses. The only instances\n    of private subclasses are the X, Y, or Z Pauli gates defined below.\n    \"\"\"\n    _XYZ = None  # type: Tuple[Pauli, Pauli, Pauli]\n\n    @staticmethod\n    def by_index(index: int) -> 'Pauli':\n        return Pauli._XYZ[index % 3]\n\n    @staticmethod\n    def by_relative_index(p: 'Pauli', relative_index: int) -> 'Pauli':\n        return Pauli._XYZ[(p._index + relative_index) % 3]\n\n    def __init__(self, index: int, name: str) -> None:\n        self._index = index\n        self._name = name\n\n    def num_qubits(self):\n        return 1\n\n    def _commutes_(self, other: Any,\n                   atol: float) -> Union[bool, NotImplementedType, None]:\n        if not isinstance(other, Pauli):\n            return NotImplemented\n        return self is other\n\n    def third(self, second: 'Pauli') -> 'Pauli':\n        return Pauli._XYZ[(-self._index - second._index) % 3]\n\n    def relative_index(self, second: 'Pauli') -> int:\n        \"\"\"Relative index of self w.r.t. second in the (X, Y, Z) cycle.\"\"\"\n        return (self._index - second._index + 1) % 3 - 1\n\n    def phased_pauli_product(\n            self, other: Union['cirq.Pauli', 'identity.IdentityGate']\n    ) -> Tuple[complex, Union['cirq.Pauli', 'identity.IdentityGate']]:\n        if self == other:\n            return 1, identity.I\n        if other is identity.I:\n            return 1, self\n        return 1j**cast(Pauli, other).relative_index(self), self.third(\n            cast(Pauli, other))\n\n    def __gt__(self, other):\n        if not isinstance(other, Pauli):\n            return NotImplemented\n        return (self._index - other._index) % 3 == 1\n\n    def __lt__(self, other):\n        if not isinstance(other, Pauli):\n            return NotImplemented\n        return (other._index - self._index) % 3 == 1\n\n    def on(self, *qubits: 'cirq.Qid') -> 'SingleQubitPauliStringGateOperation':\n        \"\"\"Returns an application of this gate to the given qubits.\n\n        Args:\n            *qubits: The collection of qubits to potentially apply the gate to.\n        \"\"\"\n        if len(qubits) != 1:\n            raise ValueError(\n                'Expected a single qubit, got <{!r}>.'.format(qubits))\n        from cirq.ops.pauli_string import SingleQubitPauliStringGateOperation\n        return SingleQubitPauliStringGateOperation(self, qubits[0])\n\n    @property\n    def _canonical_exponent(self):\n        \"\"\"Overrides EigenGate._canonical_exponent in subclasses.\"\"\"\n        return 1\n\n\nclass _PauliX(Pauli, common_gates.XPowGate):\n\n    def __init__(self):\n        Pauli.__init__(self, index=0, name='X')\n        common_gates.XPowGate.__init__(self, exponent=1.0)\n\n    def __pow__(self: '_PauliX',\n                exponent: value.TParamVal) -> common_gates.XPowGate:\n        return common_gates.XPowGate(exponent=exponent)\n\n    def _with_exponent(self: '_PauliX',\n                       exponent: value.TParamVal) -> common_gates.XPowGate:\n        return self.__pow__(exponent)\n\n    @classmethod\n    def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n        assert global_shift == 0\n        assert exponent == 1\n        return Pauli._XYZ[0]\n\n\nclass _PauliY(Pauli, common_gates.YPowGate):\n\n    def __init__(self):\n        Pauli.__init__(self, index=1, name='Y')\n        common_gates.YPowGate.__init__(self, exponent=1.0)\n\n    def __pow__(self: '_PauliY',\n                exponent: value.TParamVal) -> common_gates.YPowGate:\n        return common_gates.YPowGate(exponent=exponent)\n\n    def _with_exponent(self: '_PauliY',\n                       exponent: value.TParamVal) -> common_gates.YPowGate:\n        return self.__pow__(exponent)\n\n    @classmethod\n    def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n        assert global_shift == 0\n        assert exponent == 1\n        return Pauli._XYZ[1]\n\n\nclass _PauliZ(Pauli, common_gates.ZPowGate):\n\n    def __init__(self):\n        Pauli.__init__(self, index=2, name='Z')\n        common_gates.ZPowGate.__init__(self, exponent=1.0)\n\n    def __pow__(self: '_PauliZ',\n                exponent: value.TParamVal) -> common_gates.ZPowGate:\n        return common_gates.ZPowGate(exponent=exponent)\n\n    def _with_exponent(self: '_PauliZ',\n                       exponent: value.TParamVal) -> common_gates.ZPowGate:\n        return self.__pow__(exponent)\n\n    @classmethod\n    def _from_json_dict_(cls, exponent, global_shift, **kwargs):\n        assert global_shift == 0\n        assert exponent == 1\n        return Pauli._XYZ[2]\n\n\nX = _PauliX()\ndocument(\n    X, \"\"\"The Pauli X gate.\n\n    Matrix:\n\n        [[0, 1],\n         [1, 0]]\n    \"\"\")\n\nY = _PauliY()\ndocument(\n    Y, \"\"\"The Pauli Y gate.\n\n    Matrix:\n\n        [[0, -i],\n         [i, 0]]\n    \"\"\")\n\nZ = _PauliZ()\ndocument(\n    Z, \"\"\"The Pauli Z gate.\n\n    Matrix:\n\n        [[1, 0],\n         [0, -1]]\n    \"\"\")\n\nPauli._XYZ = (X, Y, Z)\n", "path": "cirq/ops/pauli_gates.py"}]}