Fixing compiler rule

[andrea-pasquale]

I believe that this was the issue which was giving us a weird behavior in the single qubit tomography @stavros11 @alecandido.

If we consider the definition of GPI2 provided by Qibo we have that $GPI2(0) = R_X(\pi/2)$ and $GPI2(\pi /2) = R_Y(\pi/2)$ which means that the GPI2 should have the following representation

$$ GPI2(\phi) = e^{-i \frac{\pi}{2} ( \cos{\phi} X + \sin{\phi} Y)} $$

We now need to compute how $GPI2(\phi)$ changes when we apply an $R_Z(\tilde \phi)$ gate

$$ R_Z(\tilde \phi) GPI2(\phi) R_Z(- \tilde \phi) = GPI2( \phi^*) $$

We can find $\phi^*$ with the following passages

$$ R_Z ( \tilde \phi) e^{-i \frac{\pi}{2} ( \cos{\phi} X + \sin{\phi} Y)} R_Z (- \tilde \phi) = e^{-i \frac{\pi}{2} ( \cos{(\phi - \tilde \phi)} X + \sin{(\phi - \tilde \phi)} Y)} =GPI2 (\phi - \tilde \phi). $$

Therefore $\phi^* = \phi - \tilde \phi$ which means that rule for $R_Z(\tilde \phi)$ should be $\phi \rightarrow \phi - \tilde \phi$

@stavros11 feel free to check and let me know if the fix works for you as well.
Currently this PR is pointing to main feel free to merge it directly to 0.2 or 0.1.

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 70.24%. Comparing base (9bbbbe8) to head (cb04076).

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##             main    #1044   +/-   ##
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  Coverage   70.24%   70.24%           
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  Files          64       64           
  Lines        6701     6701           
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  Hits         4707     4707           
  Misses       1994     1994           

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[alecandido]

Thanks @andrea-pasquale. I agree with the explanation, and I trust your experiment.

It would be nice to have this fix in both 0.1 and 0.2 (e.g. we could merge this PR in 0.1, and I could repeat the fix in 0.2). But before merging, I'd like to have a test failing in main and passing here to merge together, in order to avoid regressions.

If no one will volunteer before, I'll provide the test myself.

I have realized it is not ready for review

[Edoardo-Pedicillo]

There is a small typo in the last formula ( it should be $R_Z ( - \tilde \phi) e^{-i \frac{\pi}{2} ( \cos{\phi} X + \sin{\phi} Y)} R_Z ( \tilde \phi)$, so $\phi \rightarrow \phi + \tilde \phi$ ) .

Btw, from the Manenti book, sec 14.6.1, we know that a generic sinusoidal pulse $V(t)=A\epsilon(t) sin(\omega t + \alpha)$ corresponds to a generic rotation

$$ R_{n(\alpha)} (\theta) = e^{-i \frac{\theta}{2} ( \cos{(\alpha)} X + \sin{(\alpha)} Y)} $$

and it holds that (sec.14.6.4)

$$ Z_{-\alpha} X_{\beta} Z_{\alpha} = R_{n(-\alpha)} (\beta). $$

Since we can decompose a generic GPI2 as

$$ GPI2(\phi) = Z_{\phi} X_{\pi/2} Z_{-\phi} = R_{n(\phi)} (\pi/2), $$

we can see that the sign of the GPI2 phase in qibolab is correct and we have to change the Z rule, in principle leaving the latter as it is and changing the sign of the phase in the $GPI2 (\phi)$ should give the same results when we execute it, but the error will propagate when we execute 2q gates.

[alecandido]

I didn't take the time yet to check it yet. But if you're using it, I'm open to trust your results.

To properly determine the sign, it should be traced in the calculation (as you did), and through the whole usage in the code.
In any case, there are just two options, and if one is proven to work, while the other not, it's quite likely it is the correct answer (though there could be two errors balancing only in the tested case, but failing somewhere else - but it's not that complex, so there isn't that much room for something like that).

Whenever it will be merged, I will apply the same fix to 0.2.

[alecandido]

@stavros11 maybe this is not the final solution to all the (default) compiler problems, but it's well-documented (and arguably well-motivated), and it fixes something.

I'd propose to merge this and the companion PR #1057 for 0.2, and then scheduler a deeper review of the transpiler/compiler.