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]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.99%. Comparing base (d8cc501) to head (8513ff0).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1399   +/-   ##
=======================================
  Coverage   99.99%   99.99%           
=======================================
  Files          78       78           
  Lines       11192    11195    +3     
=======================================
+ Hits        11191    11194    +3     
  Misses          1        1           

Flag	Coverage Δ
unittests	99.99% <100.00%> (ø)

Flags with carried forward coverage won't be shown. Click here to find out more.

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[renatomello]

Before merging: did anyone check if the GPI2 is actually a Clifford in those parameters?

[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. #1399 (comment)