Override gate.controlled() for GlobalPhaseGate to return a ZPowGate (#6073)

* Override gate.controlled() for GlobalPhaseGate to return a ZPowGate

* Test unitary equivalence

* Override controlled only if gate is not parameterized

* Fix typo

* Fix type check

* another attempt at fixing types

* Add a comment and additional tests

Author: tanujkhattar

Diff for: cirq-core/cirq/ops/global_phase_op.py

@@ -13,14 +13,14 @@
 # limitations under the License.
 """A no-qubit global phase operation."""
 
-from typing import AbstractSet, Any, cast, Dict, Sequence, Tuple, Union
+from typing import AbstractSet, Any, cast, Dict, Sequence, Tuple, Union, Optional, Collection
 
 import numpy as np
 import sympy
 
 import cirq
 from cirq import value, protocols
-from cirq.ops import raw_types
+from cirq.ops import raw_types, controlled_gate, control_values as cv
 from cirq.type_workarounds import NotImplementedType
 
 
@@ -91,6 +91,32 @@ def _resolve_parameters_(
         coefficient = protocols.resolve_parameters(self.coefficient, resolver, recursive)
         return GlobalPhaseGate(coefficient=coefficient)
 
+    def controlled(
+        self,
+        num_controls: Optional[int] = None,
+        control_values: Optional[
+            Union[cv.AbstractControlValues, Sequence[Union[int, Collection[int]]]]
+        ] = None,
+        control_qid_shape: Optional[Tuple[int, ...]] = None,
+    ) -> raw_types.Gate:
+        result = super().controlled(num_controls, control_values, control_qid_shape)
+        if (
+            not self._is_parameterized_()
+            and isinstance(result, controlled_gate.ControlledGate)
+            and isinstance(result.control_values, cv.ProductOfSums)
+            and result.control_values[-1] == (1,)
+            and result.control_qid_shape[-1] == 2
+        ):
+            # A `GlobalPhaseGate` controlled on a qubit in state `|1>` is equivalent
+            # to applying a `ZPowGate`. This override ensures that `global_phase_gate.controlled()`
+            # returns a `ZPowGate` instead of a `ControlledGate(sub_gate=global_phase_gate)`.
+            coefficient = complex(self.coefficient)
+            exponent = float(np.angle(coefficient) / np.pi)
+            return cirq.ZPowGate(exponent=exponent).controlled(
+                result.num_controls() - 1, result.control_values[:-1], result.control_qid_shape[:-1]
+            )
+        return result
+
 
 def global_phase_operation(
     coefficient: 'cirq.TParamValComplex', atol: float = 1e-8

Diff for: cirq-core/cirq/ops/global_phase_op_test.py

@@ -279,3 +279,24 @@ def test_resolve_error(resolve_fn):
     gpt = cirq.GlobalPhaseGate(coefficient=t)
     with pytest.raises(ValueError, match='Coefficient is not unitary'):
         resolve_fn(gpt, {'t': -2})
+
+
+@pytest.mark.parametrize(
+    'coeff, exp', [(-1, 1), (1j, 0.5), (-1j, -0.5), (1 / np.sqrt(2) * (1 + 1j), 0.25)]
+)
+def test_global_phase_gate_controlled(coeff, exp):
+    g = cirq.GlobalPhaseGate(coeff)
+    op = cirq.global_phase_operation(coeff)
+    q = cirq.LineQubit.range(3)
+    for num_controls, target_gate in zip(range(1, 4), [cirq.Z, cirq.CZ, cirq.CCZ]):
+        assert g.controlled(num_controls) == target_gate**exp
+        np.testing.assert_allclose(
+            cirq.unitary(cirq.ControlledGate(g, num_controls)),
+            cirq.unitary(g.controlled(num_controls)),
+        )
+        assert op.controlled_by(*q[:num_controls]) == target_gate(*q[:num_controls]) ** exp
+    assert g.controlled(control_values=[0]) == cirq.ControlledGate(g, control_values=[0])
+    xor_control_values = cirq.SumOfProducts(((0, 0), (1, 1)))
+    assert g.controlled(control_values=xor_control_values) == cirq.ControlledGate(
+        g, control_values=xor_control_values
+    )