Adjusting Clifford condition in `gates.GPI2`

[MatteoRobbiati]

Checklist:

• [ ] Reviewers confirm new code works as expected.
• [ ] Tests are passing.
• [ ] Coverage does not decrease.
• [ ] Documentation is updated.

[codecov[bot]]

Codecov Report

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Additional details and impacted files

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[AlejandroSopena]

Yes, it is because $GPI2(0)=\sigma_x$, $GPI2(\pi/2)=\sigma_y$, $GPI2(\pi)=-\sigma_x$ and $GPI2(3\pi/2)=-\sigma_y$, where $\sigma_j$ are the Pauli matrices.

[renatomello]

Yes, it is because GPI2(0)=σx, GPI2(π/2)=σy, GPI2(π)=−σx and GPI2(3π/2)=−σy, where σj are the Pauli matrices.

I don't see it from the $GPI2$ matrix representation.

[AlejandroSopena]

Sorry, you are right. This happens for the GPI gate but not for GPI2.
It can be checked that it is a Clifford with the following code:

paulis = [gates.X(0).matrix(), gates.Y(0).matrix(), gates.Z(0).matrix()]
for p in paulis:
    for t in [0, np.pi/2, np.pi, 3*np.pi/2, 2*np.pi]:
        gate = gates.GPI2(0, t).matrix()
        gate1 = gate@p@gate.transpose().conj()
        print(gate1)

[alecandido]

That's an error: GPI2 is always a rotation by $\pi/2$, not $\pi$. So, they are square roots of Pauli's.

E.g., cf. https://ionq.com/docs/getting-started-with-native-gates#gpi2

[AlejandroSopena]

In the previous code I was printing gate instead of gate1. Now it can be seen that it maps paulis to paulis up to phases.

[alecandido]

@AlejandroSopena of course, my comment was referred not to the one directly above, but to the previous one, i.e. https://github.com/qiboteam/qibo/pull/1399#issuecomment-2255730701