Cz calibration
For each working point: (frequency, duration) we perform a cz calibration phase sweep.
The sweep is done first with the Control qubit ON (an X gate is applied on the control qubit),
producing a sinusoidal oscillation of the state 0 probability in terms of the ramsey phases.
The sweep is repeated with the Control qubit OFF,
producing a shifted sinusoidal oscillation of the state 0 probability in terms of the ramsey phases.
The (frequency, duration) working point where the phase shift is closest to \(180^o\) is kept as the Quantity of Interest values.
A derivation of the phase shift follows:
Definitions:
\(X\left|0\right\rangle =\left|1\right\rangle\)
\(X90\left|0\right\rangle = \frac{1}{\sqrt{2}}\left(\left|0\right\rangle -\imath\left|1\right\rangle \right)\)
\(X90\left|1\right\rangle = \frac{1}{\sqrt{2}}\left(-\imath\left|0\right\rangle +\left|1\right\rangle \right)\)
\(X90_{\phi}\left|0\right\rangle = \left|0\right\rangle -\imath e^{\imath\phi}\left|1\right\rangle\)
\(X90_{\phi}\left|1\right\rangle = -\imath e^{-\imath\phi}\left|0\right\rangle +\left|1\right\rangle\)
The Control Phase gate acts as:
\(CPh\left|0\right\rangle \left|0\right\rangle =\left|0\right\rangle \left|0\right\rangle\)
\(CPh\left|0\right\rangle \left|1\right\rangle =\left|0\right\rangle \left|1\right\rangle\)
\(CPh\left|1\right\rangle \left|0\right\rangle =\left|1\right\rangle \left|0\right\rangle\)
\(CPh\left|1\right\rangle \left|1\right\rangle =e^{\imath\delta}\left|1\right\rangle \left|1\right\rangle\)
Control ON
First operation apply X on control and X90 on target:
\[\begin{align*}
X\left|0\right\rangle X90\left|0\right\rangle & =\frac{1}{\sqrt{2}}\left|1\right\rangle \left(\left|0\right\rangle -\imath\left|1\right\rangle \right)\\
& =\left|1\right\rangle \left|0\right\rangle -\imath\left|1\right\rangle \left|1\right\rangle
\end{align*}\]
Second operation apply the Controled Phase:
\[CPh\left(\left|1\right\rangle \left|0\right\rangle -\imath\left|1\right\rangle \left|1\right\rangle \right)=\left(\left|1\right\rangle \left|0\right\rangle -\imath e^{\imath\delta}\left|1\right\rangle \left|1\right\rangle \right)\]
Third operation apply X on control and sweep ramsey phase on target:
\[\begin{align*}
X^{c}X90_{\phi}^{t}\left(\left|1\right\rangle \left|0\right\rangle -\imath e^{\imath\delta}\left|1\right\rangle \left|1\right\rangle \right) & =X\left|1\right\rangle X90_{\phi}\left|0\right\rangle -\imath e^{\imath\delta} X\left|1\right\rangle X90_{\phi}\left|1\right\rangle \\
& =\left(1-e^{\imath\delta-\imath\phi}\right)\left|0\right\rangle \left|0\right\rangle -\imath e^{\imath\delta}\left(1+e^{\imath\phi-\imath\delta}\right)\left|0\right\rangle \left|1\right\rangle
\end{align*}\]
The probability of the target being at the \left|1\right\rangle is the squared amplitude of the second term:
\[\begin{align*}
p_{target:1} & =\left(1+e^{\imath\left(\delta-\phi\right)}\right)\left(1+e^{-\imath\left(\delta-\phi\right)}\right)=2+e^{\imath\left(\delta-\phi\right)}+e^{-\imath\left(\delta-\phi\right)}\\
& =1/2+e^{\imath\left(\delta-\phi\right)}-e^{-\imath\left(\delta-\phi\right)}=\boxed{1/2(1+\cos\left(\delta-\phi\right))}
\end{align*}\]
Control OFF
\[\begin{align*}
\left|0\right\rangle X90\left|0\right\rangle & =\frac{1}{\sqrt{2}}\left|0\right\rangle \left(\left|0\right\rangle -\imath\left|1\right\rangle \right)\\
& =\left|0\right\rangle \left|0\right\rangle -\imath\left|0\right\rangle \left|1\right\rangle
\end{align*}\]
\[CPh\left(\left|0\right\rangle \left|0\right\rangle -\imath\left|0\right\rangle \left|1\right\rangle \right)=\left|0\right\rangle \left|0\right\rangle -\imath\left|0\right\rangle \left|1\right\rangle\]
\[\begin{align*}
X90_{\phi}^{t}\left(\left|0\right\rangle \left|0\right\rangle -\imath\left|0\right\rangle \left|1\right\rangle \right) & =\left(\left|0\right\rangle X90_{\phi}\left|0\right\rangle -\imath\left|0\right\rangle X90_{\phi}\left|1\right\rangle \right)\\
& =\left[\left(1-e^{-\imath\phi}\right)\left|0\right\rangle \left|0\right\rangle -\imath\left(1+e^{\imath\phi}\right)\left|0\right\rangle \left|1\right\rangle \right]
\end{align*}\]
\[\begin{align*}
p_{target:1} & =1/2\left(1+e^{-\imath\phi}\right)\left(1+e^{+\imath\phi}\right)\\
& =1/2+e^{-\imath\phi}+e^{+\imath\phi}=1/2+\cos{\left(\phi\right)}\\
& =\boxed{1/2(1+\cos{\left(\phi\right))}}
\end{align*}\]
Complaring the expressions of the sinusoidal oscillations in terms of the Ramsey phase \(\phi\),
they differ by the phase \(\delta\) caused by the Control Phase gate.
If this \(\delta\) is equal to \(180^{\circ}\) this operation implements a CZ gate.