Change merge_single_qubit_moments_to_phxz to support global phase. (#7405)

This addresses an issue I found while working on #7291 

Currently `merge_single_qubit_moments_to_phxz` will not merge moments
that have a global phase, meaning that moments that could theoretically
be merged are not. I updated it so that global phase is also supported.
I added two tests to show the issue.

Perhaps this situation is not common right now, but if #7383 is
committed and the flag is used, it may become more common. @daxfohl

Note that I'm not merging the global phase operations in a single one,
because I'm thinking some phases could be parametrized; should I attempt
to do this?

Author: codrut3

cirq-core/cirq/transformers/merge_single_qubit_gates.py

@@ -118,7 +118,7 @@ def merge_single_qubit_moments_to_phxz(
 
     def can_merge_moment(m: cirq.Moment):
         return all(
-            protocols.num_qubits(op) == 1
+            protocols.num_qubits(op) <= 1
             and protocols.has_unitary(op)
             and tags_to_ignore.isdisjoint(op.tags)
             for op in m
@@ -146,6 +146,10 @@ def merge_func(m1: cirq.Moment, m2: cirq.Moment) -> cirq.Moment | None:
                     )
                     if gate:
                         ret_ops.append(gate(q))
+        # Transfer global phase
+        for op in m1.operations + m2.operations:
+            if protocols.num_qubits(op) == 0:
+                ret_ops.append(op)
         return circuits.Moment(ret_ops)
 
     return transformer_primitives.merge_moments(

cirq-core/cirq/transformers/merge_single_qubit_gates_test.py

@@ -231,3 +231,35 @@ def test_merge_single_qubit_gates_to_phased_x_and_z_global_phase():
     c = cirq.Circuit(cirq.GlobalPhaseGate(1j).on())
     c2 = cirq.merge_single_qubit_gates_to_phased_x_and_z(c)
     assert c == c2
+
+
+def test_merge_single_qubit_moments_to_phxz_with_global_phase_in_first_moment():
+    q0 = cirq.LineQubit(0)
+    c_orig = cirq.Circuit(
+        cirq.Moment(cirq.Y(q0) ** 0.5, cirq.GlobalPhaseGate(1j**0.5).on()), cirq.Moment(cirq.X(q0))
+    )
+    c_expected = cirq.Circuit(
+        cirq.Moment(
+            cirq.PhasedXZGate(axis_phase_exponent=-0.5, x_exponent=0.5, z_exponent=-1.0).on(q0),
+            cirq.GlobalPhaseGate(1j**0.5).on(),
+        )
+    )
+    context = cirq.TransformerContext(tags_to_ignore=["ignore"])
+    c_new = cirq.merge_single_qubit_moments_to_phxz(c_orig, context=context)
+    assert c_new == c_expected
+
+
+def test_merge_single_qubit_moments_to_phxz_with_global_phase_in_second_moment():
+    q0 = cirq.LineQubit(0)
+    c_orig = cirq.Circuit(
+        cirq.Moment(cirq.Y(q0) ** 0.5), cirq.Moment(cirq.X(q0), cirq.GlobalPhaseGate(1j**0.5).on())
+    )
+    c_expected = cirq.Circuit(
+        cirq.Moment(
+            cirq.PhasedXZGate(axis_phase_exponent=-0.5, x_exponent=0.5, z_exponent=-1.0).on(q0),
+            cirq.GlobalPhaseGate(1j**0.5).on(),
+        )
+    )
+    context = cirq.TransformerContext(tags_to_ignore=["ignore"])
+    c_new = cirq.merge_single_qubit_moments_to_phxz(c_orig, context=context)
+    assert c_new == c_expected