diff --git a/cirq/ops/common_gates_test.py b/cirq/ops/common_gates_test.py
index f1d4950e874..e6deab595ed 100644
--- a/cirq/ops/common_gates_test.py
+++ b/cirq/ops/common_gates_test.py
@@ -575,7 +575,12 @@ def test_act_on_ch_form(input_gate_sequence, outcome):
     else:
         assert num_qubits == 2
         axes = [0, 1]
-    args = cirq.ActOnStabilizerCHFormArgs(state=original_state.copy(), axes=axes)
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=original_state.copy(),
+        axes=axes,
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
 
     flipped_state = cirq.StabilizerStateChForm(num_qubits=5, initial_state=23)
 
diff --git a/cirq/ops/global_phase_op_test.py b/cirq/ops/global_phase_op_test.py
index dd310c6252e..9468c6c0151 100644
--- a/cirq/ops/global_phase_op_test.py
+++ b/cirq/ops/global_phase_op_test.py
@@ -50,7 +50,12 @@ def test_act_on_tableau(phase):
 @pytest.mark.parametrize('phase', [1, 1j, -1])
 def test_act_on_ch_form(phase):
     state = cirq.StabilizerStateChForm(0)
-    args = cirq.ActOnStabilizerCHFormArgs(state, [])
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state,
+        [],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
     cirq.act_on(cirq.GlobalPhaseOperation(phase), args, allow_decompose=False)
     assert state.state_vector() == [[phase]]
 
diff --git a/cirq/ops/measurement_gate.py b/cirq/ops/measurement_gate.py
index f553fdd7dff..a8dfc29293e 100644
--- a/cirq/ops/measurement_gate.py
+++ b/cirq/ops/measurement_gate.py
@@ -242,6 +242,13 @@ def _act_on_(self, args: Any) -> bool:
             args.record_measurement_result(self.key, corrected)
             return True
 
+        if isinstance(args, sim.clifford.ActOnStabilizerCHFormArgs):
+            invert_mask = self.full_invert_mask()
+            bits = [args.state._measure(q, args.prng) for q in args.axes]
+            corrected = [bit ^ (bit < 2 and mask) for bit, mask in zip(bits, invert_mask)]
+            args.record_measurement_result(self.key, corrected)
+            return True
+
         return NotImplemented
 
 
diff --git a/cirq/ops/measurement_gate_test.py b/cirq/ops/measurement_gate_test.py
index 223979e6074..a3c7fda2d5d 100644
--- a/cirq/ops/measurement_gate_test.py
+++ b/cirq/ops/measurement_gate_test.py
@@ -344,6 +344,47 @@ def test_act_on_clifford_tableau():
         cirq.act_on(m, args)
 
 
+def test_act_on_stabilizer_ch_form():
+    a, b = cirq.LineQubit.range(2)
+    m = cirq.measure(a, b, key='out', invert_mask=(True,))
+    # The below assertion does not fail since it ignores non-unitary operations
+    cirq.testing.assert_all_implemented_act_on_effects_match_unitary(m)
+
+    with pytest.raises(TypeError, match="Failed to act"):
+        cirq.act_on(m, object())
+
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=cirq.StabilizerStateChForm(num_qubits=5, initial_state=0),
+        axes=[3, 1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
+    cirq.act_on(m, args)
+    assert args.log_of_measurement_results == {'out': [1, 0]}
+
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=cirq.StabilizerStateChForm(num_qubits=5, initial_state=8),
+        axes=[3, 1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
+
+    cirq.act_on(m, args)
+    assert args.log_of_measurement_results == {'out': [1, 1]}
+
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=cirq.StabilizerStateChForm(num_qubits=5, initial_state=10),
+        axes=[3, 1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
+    cirq.act_on(m, args)
+    assert args.log_of_measurement_results == {'out': [0, 1]}
+
+    with pytest.raises(ValueError, match="already logged to key"):
+        cirq.act_on(m, args)
+
+
 def test_act_on_qutrit():
     a, b = cirq.LineQid.range(2, dimension=3)
     m = cirq.measure(a, b, key='out', invert_mask=(True,))
diff --git a/cirq/sim/clifford/act_on_stabilizer_ch_form_args.py b/cirq/sim/clifford/act_on_stabilizer_ch_form_args.py
index 450a3743970..8d9ec2a868b 100644
--- a/cirq/sim/clifford/act_on_stabilizer_ch_form_args.py
+++ b/cirq/sim/clifford/act_on_stabilizer_ch_form_args.py
@@ -12,7 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-from typing import Any, Iterable, TYPE_CHECKING
+from typing import Any, Dict, Iterable, TYPE_CHECKING
 
 import numpy as np
 
@@ -31,19 +31,43 @@ class ActOnStabilizerCHFormArgs:
 
     To act on this object, directly edit the `state` property, which is
     storing the stabilizer state of the quantum system with one axis per qubit.
-    Measurements are currently not supported on this object.
     """
 
-    def __init__(self, state: StabilizerStateChForm, axes: Iterable[int]):
+    def __init__(
+        self,
+        state: StabilizerStateChForm,
+        axes: Iterable[int],
+        prng: np.random.RandomState,
+        log_of_measurement_results: Dict[str, Any],
+    ):
         """Initializes with the given state and the axes for the operation.
         Args:
             state: The StabilizerStateChForm to act on. Operations are expected
                 to perform inplace edits of this object.
             axes: The indices of axes corresponding to the qubits that the
                 operation is supposed to act upon.
+            prng: The pseudo random number generator to use for probabilistic
+                effects.
+            log_of_measurement_results: A mutable object that measurements are
+                being recorded into. Edit it easily by calling
+                `ActOnStabilizerCHFormArgs.record_measurement_result`.
         """
         self.state = state
         self.axes = tuple(axes)
+        self.prng = prng
+        self.log_of_measurement_results = log_of_measurement_results
+
+    def record_measurement_result(self, key: str, value: Any):
+        """Adds a measurement result to the log.
+        Args:
+            key: The key the measurement result should be logged under. Note
+                that operations should only store results under keys they have
+                declared in a `_measurement_keys_` method.
+            value: The value to log for the measurement.
+        """
+        if key in self.log_of_measurement_results:
+            raise ValueError(f"Measurement already logged to key {key!r}")
+        self.log_of_measurement_results[key] = value
 
     def _act_on_fallback_(self, action: Any, allow_decompose: bool):
         strats = []
diff --git a/cirq/sim/clifford/act_on_stabilizer_ch_form_args_test.py b/cirq/sim/clifford/act_on_stabilizer_ch_form_args_test.py
index f85c5840aa0..8f3963cd4ce 100644
--- a/cirq/sim/clifford/act_on_stabilizer_ch_form_args_test.py
+++ b/cirq/sim/clifford/act_on_stabilizer_ch_form_args_test.py
@@ -22,7 +22,12 @@ def test_cannot_act():
     class NoDetails(cirq.SingleQubitGate):
         pass
 
-    args = cirq.ActOnStabilizerCHFormArgs(state=cirq.StabilizerStateChForm(num_qubits=3), axes=[1])
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=cirq.StabilizerStateChForm(num_qubits=3),
+        axes=[1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
 
     with pytest.raises(TypeError, match="Failed to act"):
         cirq.act_on(NoDetails(), args)
@@ -37,7 +42,12 @@ def _act_on_(self, args):
                 return True
 
     state = cirq.StabilizerStateChForm(num_qubits=3)
-    args = cirq.ActOnStabilizerCHFormArgs(state=state, axes=[1])
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=state,
+        axes=[1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
 
     cirq.act_on(CustomGate(), args)
 
@@ -54,9 +64,19 @@ def _unitary_(self):
 
     original_state = cirq.StabilizerStateChForm(num_qubits=3)
 
-    args = cirq.ActOnStabilizerCHFormArgs(state=original_state.copy(), axes=[1])
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=original_state.copy(),
+        axes=[1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
     cirq.act_on(UnitaryYGate(), args)
-    expected_args = cirq.ActOnStabilizerCHFormArgs(state=original_state.copy(), axes=[1])
+    expected_args = cirq.ActOnStabilizerCHFormArgs(
+        state=original_state.copy(),
+        axes=[1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
     cirq.act_on(cirq.Y, expected_args)
     np.testing.assert_allclose(args.state.state_vector(), expected_args.state.state_vector())
 
@@ -71,8 +91,18 @@ def _unitary_(self):
 
     original_state = cirq.StabilizerStateChForm(num_qubits=3)
 
-    args = cirq.ActOnStabilizerCHFormArgs(state=original_state.copy(), axes=[1])
+    args = cirq.ActOnStabilizerCHFormArgs(
+        state=original_state.copy(),
+        axes=[1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
     cirq.act_on(UnitaryHGate(), args)
-    expected_args = cirq.ActOnStabilizerCHFormArgs(state=original_state.copy(), axes=[1])
+    expected_args = cirq.ActOnStabilizerCHFormArgs(
+        state=original_state.copy(),
+        axes=[1],
+        prng=np.random.RandomState(),
+        log_of_measurement_results={},
+    )
     cirq.act_on(cirq.H, expected_args)
     np.testing.assert_allclose(args.state.state_vector(), expected_args.state.state_vector())
diff --git a/cirq/sim/clifford/clifford_simulator.py b/cirq/sim/clifford/clifford_simulator.py
index f38c0e6d179..549a77c8edb 100644
--- a/cirq/sim/clifford/clifford_simulator.py
+++ b/cirq/sim/clifford/clifford_simulator.py
@@ -335,7 +335,7 @@ def apply_unitary(self, op: 'cirq.Operation'):
             self.tableau, [self.qubit_map[i] for i in op.qubits], np.random.RandomState(), {}
         )
         ch_form_args = clifford.ActOnStabilizerCHFormArgs(
-            self.ch_form, [self.qubit_map[i] for i in op.qubits]
+            self.ch_form, [self.qubit_map[i] for i in op.qubits], np.random.RandomState(), {}
         )
         try:
             act_on(op, tableau_args)
diff --git a/cirq/sim/clifford/stabilizer_state_ch_form.py b/cirq/sim/clifford/stabilizer_state_ch_form.py
index 7e1b8ba8bcd..219dedf5a02 100644
--- a/cirq/sim/clifford/stabilizer_state_ch_form.py
+++ b/cirq/sim/clifford/stabilizer_state_ch_form.py
@@ -61,7 +61,10 @@ def __init__(self, num_qubits: int, initial_state: int = 0) -> None:
             big_endian_int_to_digits(initial_state, digit_count=num_qubits, base=2)
         ):
             if val:
-                protocols.act_on(pauli_gates.X, clifford.ActOnStabilizerCHFormArgs(self, [i]))
+                protocols.act_on(
+                    pauli_gates.X,
+                    clifford.ActOnStabilizerCHFormArgs(self, [i], np.random.RandomState(), {}),
+                )
 
     def _json_dict_(self) -> Dict[str, Any]:
         return protocols.obj_to_dict_helper(self, ['n', 'G', 'F', 'M', 'gamma', 'v', 's', 'omega'])
diff --git a/cirq/testing/consistent_act_on.py b/cirq/testing/consistent_act_on.py
index eaf1890d99b..76a3a2f5813 100644
--- a/cirq/testing/consistent_act_on.py
+++ b/cirq/testing/consistent_act_on.py
@@ -190,7 +190,10 @@ def _final_stabilizer_state_ch_form(
     for op in circuit.all_operations():
         try:
             args = act_on_stabilizer_ch_form_args.ActOnStabilizerCHFormArgs(
-                state=stabilizer_ch_form, axes=[qubit_map[qid] for qid in op.qubits]
+                state=stabilizer_ch_form,
+                axes=[qubit_map[qid] for qid in op.qubits],
+                prng=np.random.RandomState(),
+                log_of_measurement_results={},
             )
             protocols.act_on(op, args, allow_decompose=True)
         except TypeError: