diff --git a/cirq/circuits/circuit_test.py b/cirq/circuits/circuit_test.py
index 2b420247ef8..95c1b3941cc 100644
--- a/cirq/circuits/circuit_test.py
+++ b/cirq/circuits/circuit_test.py
@@ -14,7 +14,7 @@
 import os
 from collections import defaultdict
 from random import randint, random, sample, randrange
-from typing import Tuple
+from typing import Optional, Tuple, TYPE_CHECKING
 
 import numpy as np
 import pytest
@@ -65,6 +65,10 @@ def can_add_operation_into_moment(
 )
 
 
+if TYPE_CHECKING:
+    import cirq
+
+
 class _MomentAndOpTypeValidatingDeviceType(cirq.Device):
     def validate_operation(self, operation):
         if not isinstance(operation, cirq.Operation):
@@ -958,7 +962,9 @@ def __init__(self, replacer=(lambda x: x)):
             super().__init__()
             self.replacer = replacer
 
-        def optimization_at(self, circuit, index, op):
+        def optimization_at(
+            self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+        ) -> Optional['cirq.PointOptimizationSummary']:
             new_ops = self.replacer(op)
             return cirq.PointOptimizationSummary(
                 clear_span=1, clear_qubits=op.qubits, new_operations=new_ops
diff --git a/cirq/circuits/optimization_pass_test.py b/cirq/circuits/optimization_pass_test.py
index 1fa33dcc4cd..15184a0f149 100644
--- a/cirq/circuits/optimization_pass_test.py
+++ b/cirq/circuits/optimization_pass_test.py
@@ -11,12 +11,16 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
+from typing import Optional, TYPE_CHECKING, Set, List
 
 import pytest
 import cirq
-from cirq import PointOptimizer, PointOptimizationSummary
+from cirq import PointOptimizer, PointOptimizationSummary, Operation
 from cirq.testing import EqualsTester
 
+if TYPE_CHECKING:
+    import cirq
+
 
 def test_equality():
     a = cirq.NamedQubit('a')
@@ -57,7 +61,9 @@ class ReplaceWithXGates(PointOptimizer):
     operation's qubits.
     """
 
-    def optimization_at(self, circuit, index, op):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         end = index + 1
         new_ops = [cirq.X(q) for q in op.qubits]
         done = False
@@ -65,10 +71,10 @@ def optimization_at(self, circuit, index, op):
             n = circuit.next_moment_operating_on(op.qubits, end)
             if n is None:
                 break
-            next_ops = {circuit.operation_at(q, n) for q in op.qubits}
-            next_ops = [e for e in next_ops if e]
-            next_ops = sorted(next_ops, key=lambda e: str(e.qubits))
-            for next_op in next_ops:
+            next_ops: Set[Optional[Operation]] = {circuit.operation_at(q, n) for q in op.qubits}
+            next_ops_list: List[Operation] = [e for e in next_ops if e]
+            next_ops_sorted = sorted(next_ops_list, key=lambda e: str(e.qubits))
+            for next_op in next_ops_sorted:
                 if next_op:
                     if set(next_op.qubits).issubset(op.qubits):
                         end = n + 1
@@ -149,14 +155,19 @@ def test_point_optimizer_raises_on_gates_changing_qubits():
     class EverythingIs42(cirq.PointOptimizer):
         """Changes all single qubit operations to act on LineQubit(42)"""
 
-        def optimization_at(self, circuit, index, op):
-            if len(op.qubits) == 1:
+        def optimization_at(
+            self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+        ) -> Optional['cirq.PointOptimizationSummary']:
+            new_op = op
+            if len(op.qubits) == 1 and isinstance(op, cirq.GateOperation):
                 new_op = op.gate(cirq.LineQubit(42))
-                return cirq.PointOptimizationSummary(
-                    clear_span=1, clear_qubits=op.qubits, new_operations=new_op
-                )
+
+            return cirq.PointOptimizationSummary(
+                clear_span=1, clear_qubits=op.qubits, new_operations=new_op
+            )
 
     c = cirq.Circuit(cirq.X(cirq.LineQubit(0)), cirq.X(cirq.LineQubit(1)))
+
     with pytest.raises(ValueError, match='new qubits'):
         EverythingIs42().optimize_circuit(c)
 
diff --git a/cirq/contrib/acquaintance/executor.py b/cirq/contrib/acquaintance/executor.py
index b37be6dc63b..299ee6c347c 100644
--- a/cirq/contrib/acquaintance/executor.py
+++ b/cirq/contrib/acquaintance/executor.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-from typing import DefaultDict, Dict, Sequence, TYPE_CHECKING
+from typing import DefaultDict, Dict, Sequence, TYPE_CHECKING, Optional
 
 import abc
 from collections import defaultdict
@@ -81,7 +81,9 @@ def __call__(self, strategy: 'cirq.Circuit'):
         super().optimize_circuit(strategy)
         return self.mapping.copy()
 
-    def optimization_at(self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         if isinstance(op.gate, AcquaintanceOpportunityGate):
             logical_indices = tuple(self.mapping[q] for q in op.qubits)
             logical_operations = self.execution_strategy.get_operations(logical_indices, op.qubits)
@@ -93,7 +95,7 @@ def optimization_at(self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operati
 
         if isinstance(op, ops.GateOperation) and isinstance(op.gate, PermutationGate):
             op.gate.update_mapping(self.mapping, op.qubits)
-            return
+            return None
 
         raise TypeError(
             'Can only execute a strategy consisting of gates that '
diff --git a/cirq/google/optimizers/convert_to_sqrt_iswap.py b/cirq/google/optimizers/convert_to_sqrt_iswap.py
index 4f7d41d99ef..cf228fd3995 100644
--- a/cirq/google/optimizers/convert_to_sqrt_iswap.py
+++ b/cirq/google/optimizers/convert_to_sqrt_iswap.py
@@ -18,6 +18,7 @@
 
 from cirq import ops, circuits, protocols
 
+
 if TYPE_CHECKING:
     import cirq
 
@@ -104,7 +105,9 @@ def convert(self, op: 'cirq.Operation') -> List['cirq.Operation']:
         )
         return a
 
-    def optimization_at(self, circuit, index, op):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         if isinstance(op.gate, ops.MatrixGate) and len(op.qubits) == 1:
             return None
 
diff --git a/cirq/google/optimizers/convert_to_sycamore_gates.py b/cirq/google/optimizers/convert_to_sycamore_gates.py
index accc92e1758..225028d80de 100644
--- a/cirq/google/optimizers/convert_to_sycamore_gates.py
+++ b/cirq/google/optimizers/convert_to_sycamore_gates.py
@@ -17,6 +17,7 @@
 import numpy as np
 import scipy.linalg
 from cirq import circuits, google, linalg, ops, optimizers, protocols
+
 from cirq.google.ops import SycamoreGate
 from cirq.google.optimizers.two_qubit_gates.gate_compilation import GateTabulation
 
@@ -129,7 +130,9 @@ def on_stuck_raise(bad):
             on_stuck_raise=None if self.ignore_failures else on_stuck_raise,
         )
 
-    def optimization_at(self, circuit, index, op):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         if not isinstance(op, ops.GateOperation):
             return None
 
@@ -144,6 +147,8 @@ def optimization_at(self, circuit, index, op):
                 ops_in_front = list({circuit.operation_at(q, next_index) for q in op.qubits})
                 if len(ops_in_front) == 1 and isinstance(ops_in_front[0], ops.GateOperation):
                     gate2 = ops_in_front[0].gate
+            else:
+                next_index = 0
 
             if isinstance(gate, ops.SwapPowGate) and isinstance(gate2, ops.ZZPowGate):
                 rads = gate2.exponent * np.pi / 2
diff --git a/cirq/google/optimizers/convert_to_xmon_gates.py b/cirq/google/optimizers/convert_to_xmon_gates.py
index 85852e64391..699c91b4039 100644
--- a/cirq/google/optimizers/convert_to_xmon_gates.py
+++ b/cirq/google/optimizers/convert_to_xmon_gates.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import List, TYPE_CHECKING
+from typing import List, TYPE_CHECKING, Optional
 
 from cirq import ops, protocols
 from cirq.circuits.optimization_pass import (
@@ -90,7 +90,9 @@ def on_stuck_raise(bad):
             on_stuck_raise=None if self.ignore_failures else on_stuck_raise,
         )
 
-    def optimization_at(self, circuit, index, op):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         converted = self.convert(op)
         if len(converted) == 1 and converted[0] is op:
             return None
diff --git a/cirq/neutral_atoms/convert_to_neutral_atom_gates.py b/cirq/neutral_atoms/convert_to_neutral_atom_gates.py
index b21800888bb..cc7236fa19c 100644
--- a/cirq/neutral_atoms/convert_to_neutral_atom_gates.py
+++ b/cirq/neutral_atoms/convert_to_neutral_atom_gates.py
@@ -11,7 +11,7 @@
 # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 # See the License for the specific language governing permissions and
 # limitations under the License.
-from typing import List
+from typing import List, Optional, TYPE_CHECKING
 
 from cirq import ops, protocols
 from cirq.circuits.optimization_pass import (
@@ -20,6 +20,9 @@
 )
 from cirq import optimizers
 
+if TYPE_CHECKING:
+    import cirq
+
 
 class ConvertToNeutralAtomGates(PointOptimizer):
     """Attempts to convert gates into native Atom gates.
@@ -76,7 +79,9 @@ def on_stuck_raise(bad):
             on_stuck_raise=None if self.ignore_failures else on_stuck_raise,
         )
 
-    def optimization_at(self, circuit, index, op):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         converted = self.convert(op)
         if len(converted) == 1 and converted[0] is op:
             return None
diff --git a/cirq/optimizers/expand_composite.py b/cirq/optimizers/expand_composite.py
index 35288f32951..ba9a9ebd494 100644
--- a/cirq/optimizers/expand_composite.py
+++ b/cirq/optimizers/expand_composite.py
@@ -14,7 +14,7 @@
 
 """An optimizer that expands composite operations via `cirq.decompose`."""
 
-from typing import Callable
+from typing import Callable, Optional, TYPE_CHECKING
 
 from cirq import ops, protocols
 from cirq.circuits.optimization_pass import (
@@ -22,6 +22,9 @@
     PointOptimizationSummary,
 )
 
+if TYPE_CHECKING:
+    import cirq
+
 
 class ExpandComposite(PointOptimizer):
     """An optimizer that expands composite operations via `cirq.decompose`.
@@ -41,7 +44,9 @@ def __init__(self, no_decomp: Callable[[ops.Operation], bool] = (lambda _: False
         super().__init__()
         self.no_decomp = no_decomp
 
-    def optimization_at(self, circuit, index, op):
+    def optimization_at(
+        self, circuit: 'cirq.Circuit', index: int, op: 'cirq.Operation'
+    ) -> Optional['cirq.PointOptimizationSummary']:
         decomposition = protocols.decompose(op, keep=self.no_decomp, on_stuck_raise=None)
         if decomposition == [op]:
             return None