Cz local phases
Calibrate local phases:
Taking the local phases into account, the action of the CPhase gate can be described as:
\(CPh\left|0\right\rangle \left|0\right\rangle =\left|0\right\rangle \left|0\right\rangle\)
\(CPh\left|0\right\rangle \left|1\right\rangle =e^{-\imath\phi_{01}}\left|0\right\rangle \left|1\right\rangle\)
\(CPh\left|1\right\rangle \left|0\right\rangle =e^{-\imath\phi_{10}}\left|1\right\rangle \left|0\right\rangle\)
\(CPh\left|1\right\rangle \left|1\right\rangle =e^{-\imath\left(\phi_{01}+\phi_{10}+\phi_{g}\right)}\left|1\right\rangle \left|1\right\rangle\)
\(X90_{\phi}\left|0\right\rangle =\left|0\right\rangle -\imath e^{\imath\phi}\left|1\right\rangle\)
\(X90_{\phi}\left|1\right\rangle =-\imath e^{-\imath\phi}\left|0\right\rangle +\left|1\right\rangle\)
Target Qubit Local Phase
To calibrate the local phase for the target qubit the swap flag is set to OFF, i.e. normal roles for each qubit.
CZ ON
\[\begin{align*}
\left(X90\left|0\right\rangle \right)\left|0\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|0\right\rangle \left|0\right\rangle -\imath\left|1\right\rangle \left|0\right\rangle \right)\\
& \rightarrow\\
& \frac{1}{\sqrt{2}}Cph_{(\phi_{g})}\left(\left|0\right\rangle \left|0\right\rangle -\imath\left|1\right\rangle \left|0\right\rangle \right)=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle \left|0\right\rangle -\imath e^{-\imath\phi_{10}}\left|1\right\rangle \left|0\right\rangle \right)\\
& \rightarrow\\
& \frac{1}{\sqrt{2}}X90_{\phi}^{c}\left(\left|0\right\rangle \left|0\right\rangle -\imath e^{-\imath\phi_{10}}\left|1\right\rangle \left|0\right\rangle \right)=\left[\left(\left|0\right\rangle -\imath e^{\imath\phi}\left|1\right\rangle \right)\left|0\right\rangle -\imath e^{-\imath\phi_{10}}\left(-\imath e^{-\imath\phi}\left|0\right\rangle +\left|1\right\rangle \right)\left|0\right\rangle \right]\\
& =\left|0\right\rangle \left|0\right\rangle -\imath e^{\imath\phi}\left|1\right\rangle \left|0\right\rangle -e^{-\imath\phi}e^{-\imath\phi_{10}}\left|0\right\rangle \left|0\right\rangle -\imath e^{-\imath\phi_{10}}\left|1\right\rangle \left|0\right\rangle \\
& =\left(1-e^{-\imath\left(\phi+\phi_{10}\right)}\right)\left|0\right\rangle \left|0\right\rangle -\imath e^{\imath\phi}\left(1+e^{-\imath\left(\phi_{10}+\phi\right)}\right)\left|1\right\rangle \left|0\right\rangle \\
\\
\end{align*}\]
\[\begin{align*}
P^{1;control} & =\left(1+e^{-\imath\left(\phi_{10}+\phi\right)}\right)\left(1+e^{+\imath\left(\phi_{10}+\phi\right)}\right)\\
& =\left(1+e^{+\imath\left(\phi_{10}+\phi\right)}+e^{-\imath\left(\phi_{10}+\phi\right)}+1\right)\\
& =\boxed{\frac{1}{2}\left(1+\cos\left(\phi_{10}+\phi\right)\right)}
\end{align*}\]
CZ OFF
\[\begin{align*}
\left(X90\left|0\right\rangle \right)\left|0\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|0\right\rangle \left|0\right\rangle -\imath\left|1\right\rangle \left|0\right\rangle \right)\\
& \rightarrow\\
& \frac{1}{\sqrt{2}}\left(X90_{\phi}^{c}\left|0\right\rangle \left|0\right\rangle -\imath X90_{\phi}^{c}\left|1\right\rangle \left|0\right\rangle \right)\\
& =\frac{1}{\sqrt{2}}\left[\left(X90_{\phi}^{c}\left|0\right\rangle \right)\left|0\right\rangle -\imath\left(X90_{\phi}^{c}\left|1\right\rangle \right)\left|0\right\rangle \right]\\
& =\frac{1}{\sqrt{2}}\left[\left(\left|0\right\rangle -\imath e^{\imath\phi}\left|1\right\rangle \right)\left|0\right\rangle -\imath\left(-\imath e^{-\imath\phi}\left|0\right\rangle +\left|1\right\rangle \right)\left|0\right\rangle \right]\\
& =\frac{1}{\sqrt{2}}\left[\left|0\right\rangle \left|0\right\rangle -\imath e^{\imath\phi}\left|1\right\rangle \left|0\right\rangle -e^{-\imath\phi}\left|0\right\rangle \left|0\right\rangle -\imath\left|1\right\rangle \left|0\right\rangle \right]\\
& =\frac{1}{\sqrt{2}}\left[\left(1-e^{-\imath\phi}\right)\left|0\right\rangle \left|0\right\rangle -\imath\left(1+e^{\imath\phi}\right)\left|1\right\rangle \left|0\right\rangle \right]\\
\end{align*}\]
\[\begin{align*}
P^{1;control} & =\frac{1}{4}\left(1+e^{\imath\phi}\right)\left(1+e^{-\imath\phi}\right)=\frac{1}{2}\left(2+e^{-\imath\phi}+e^{\imath\phi}\right)\\
& =\frac{1}{4}\left(2+2\cos\phi\right)=\boxed{\frac{1}{2}\left(1+\cos\phi\right)}
\end{align*}\]
Comparing the expression for \(P^{1;control}\) the sinusoidal curves are shifted by the phase \(\phi_{10}\)
For the phase \(\phi_{01}\): Repeat with swaping control_qubit \(\leftrightarrow\) target_qubit, by setting the swap to ON
