Incorrect unitary of controlled QasmUGate?

[ArfatSalman]

Description of the issue
It seems like the unitary of controlled QasmUGate is not correct with qiskit's CU3Gate.

How to reproduce the issue

import cirq
from cirq.circuits.qasm_output import QasmUGate
import numpy as np

cu3 = QasmUGate(
    0.9284573101905567 / np.pi, 
    0.45091656513571715 / np.pi, 
    3.783886899929776 / np.pi).controlled(num_controls=1)

qr_cirq = [cirq.NamedQubit('q' + str(i)) for i in range(2)]
circuit = cirq.Circuit(
    cu3(qr_cirq[1], qr_cirq[0])
)

cirq_unitary = cirq.unitary(circuit)


import qiskit
from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister
from qiskit.circuit.library.standard_gates import CU3Gate

qr = QuantumRegister(2, name='qr')
cr = ClassicalRegister(2, name='cr')
qc = QuantumCircuit(qr, cr, name='qc')

qc.append(CU3Gate(0.9284573101905567,0.45091656513571715,3.783886899929776), qargs=[qr[1], qr[0]], cargs=[])

from qiskit import Aer, transpile, execute
backend = Aer.get_backend('unitary_simulator')
qiskit_unitary = execute(qc, backend).result().get_unitary(qc)

print(cirq.equal_up_to_global_phase(qiskit_unitary, cirq_unitary)) 
# False

As per the docs,

U3Gate corresponds to QasmUGate with half turns. So I am expecting the controlled version to correspond to CU3Gate. However, the unitaries are different for some reason.

Cirq version
1.0.0

[viathor]

Thank you for a detailed bug report! This is helpful!

Related: #5928

[tanujkhattar]

If two unitaries U1 and U2 are equal upto global phase but not identical, then their controlled versions would not be equal upto global phase because the global phase of U1/U2 becomes local phase of CU1 and CU2.

A classic example would X gate and Rx gate, both of which differ only by a global phase. For example:

>>> import cirq
>>> import numpy as np
>>> g1, g2 = cirq.X, cirq.Rx(rads=np.pi)
>>> cirq.equal_up_to_global_phase(g1, g2)
True
>>> cg1, cg2 = g1.controlled(), g2.controlled()
>>> cirq.equal_up_to_global_phase(cg1, cg2)
False
>>> cirq.unitary(cg1) # CX gate swaps 0/1 based when control is 1.
array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],
       [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j]])

>>> cirq.unitary(cg2) # CRx gate swaps 0/1 and also adds a -1 phase when control is 1. 
Out[14]:
array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
       [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j],
       [0.+0.j, 0.+0.j, 0.+0.j, 0.-1.j],
       [0.+0.j, 0.+0.j, 0.-1.j, 0.+0.j]])

Note that the two controlled gates shown in the example above are still equivalent upto single qubit rotations (eg: prepend a Z rotation on qubit 0).

[ArfatSalman]

Thank you for the detailed comment, @tanujkhattar!

[viathor]

I think @tanujkhattar is correct about the root cause. Still, the fact that qiskit's CU3Gate(...) and cirq's QasmUGate(...).controlled_by(...) are inequivalent gates sounds like a problem.