Global phase on Solovay-Kitaev decomposition seems wrong

[yutakahirano]

Environment

• Qiskit Terra version: 0.23.1
• Python version: 3.10.6
• Operating system: Ubuntu Linux

What is happening?

The global phase of some circuits generated by SolovayKitaev seems wrong.

How can we reproduce the issue?

>>> from qiskit.transpiler.passes.synthesis import SolovayKitaev
>>> from qiskit.quantum_info import Operator
>>> from qiskit import QuantumCircuit, Aer, transpile, assemble
>>> c = QuantumCircuit(1)
>>> c.y(0)
<qiskit.circuit.instructionset.InstructionSet object at 0x7f62280bc910>
>>> skd = SolovayKitaev(recursion_degree=3)
>>> dc = skd(c)
>>> c.draw()
   ┌───┐
q: ┤ Y ├
   └───┘
>>> dc.draw()
global phase: 3π/4
   ┌───┐┌───┐┌───┐┌───┐┌─────┐┌─────┐┌───┐┌───┐┌───┐
q: ┤ H ├┤ T ├┤ T ├┤ H ├┤ Tdg ├┤ Tdg ├┤ H ├┤ T ├┤ T ├
   └───┘└───┘└───┘└───┘└─────┘└─────┘└───┘└───┘└───┘
>>> Operator(c)
Operator([[0.+0.j, 0.-1.j],
          [0.+1.j, 0.+0.j]],
         input_dims=(2,), output_dims=(2,))
>>> Operator(dc)
Operator([[-1.11022302e-16+7.85046229e-17j,
            1.05981241e-15+1.00000000e+00j],
          [-9.43689571e-16-1.00000000e+00j,
            0.00000000e+00-3.92523115e-17j]],
         input_dims=(2,), output_dims=(2,))

What should happen?

Given Operator(dc).data aprroximately equals to [[0, 1j], [-1j, 0]] which equals to -Operator(c).data, dc's global phase should be π rather than 3π/4, I believe.

Any suggestions?

No response

[Cryoris]

That could probably be quick-fixed by checking the phases of the input and output matrix, but this might be a deeper bug since I think the SK algorithm should come up with the right phase in the end 🤔