[mergify]

Summary

This PR fixes a number of bugs we have recently discovered in the Python code for the Solovay-Kitaev decomposition, together with Julien Gacon (@Cryoris) and Almudena Carrera Vazquez (@anedumla).

Fixes include:

• fixed a problem with the rotation axis computation for 180 degree rotations (thanks to Julien Gacon (@Cryoris) for working out the math)
• fixed a problem where the global phase of the returned circuit might have been off by $\pi$, due to the underlying algorithm approximating not the original SU(2) matrix, but its projection onto SO(3)
• in particular fixed Global phase on Solovay-Kitaev decomposition seems wrong #9552.

Longer-term, we will deprecate the Python code for Solovay-Kitaev decomposition altogether, and switch to the new improved pass written in Rust that also fixes these problems (see #14203), however for backward compatibility we need to temporarily support the Python code as well, and thus it makes sense to fix the problems on the Python side too.

Details and comments

Other changes:

• There was a very strange "back door" for returning a circuit with the original unitary matrix if the computed matrix is not close to the identity. This behavior is not consistent with the documented description of the pass and was removed. (Or rather the functions to_circuit and to_dag remain for backwards compatibility but are no longer used by the main pass).
• The GateSequence class used to store both the SO(3) matrix as product and the SU(2) matrix as product_su2, however the parameter product_su2 was not updated correctly in various places (and it was not even clear what to do within GateSequence.from_matrix). As product_su2 was almost not used anywhere in the algorithm (with the exception of one place where its usage did not seem to be correct anyway), it was simply removed to avoid confusion and unnecessary computations.

---

This is an automatic backport of pull request #14217 done by [Mergify](https://mergify.com).

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Changes to be committed:
	modified:   qiskit/synthesis/discrete_basis/commutator_decompose.py
	modified:   qiskit/synthesis/discrete_basis/gate_sequence.py
	modified:   qiskit/synthesis/discrete_basis/generate_basis_approximations.py
	modified:   qiskit/synthesis/discrete_basis/solovay_kitaev.py
	new file:   releasenotes/notes/fix-sk-decomposition-23da3ee4b6a10d62.yaml

Unmerged paths:
  (use "git add <file>..." to mark resolution)
	both modified:   test/python/transpiler/test_solovay_kitaev.py

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