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Replace the fitter of StarkRamseyXYAmpScan experiment (#1243)
### Summary This PR replaces the fitter of `StarkRamseyXYAmpScan` experiment. Current fitter puts strong assumption on model parameter and it makes the fitting quite unstable to the change in dephasing rate. This PR doesn't introduce any breaking API change and a separate release note is not necessary because the experiment is not released yet. ### Details and comments The current fitter simultaneously fits the P1 values of the Ramsey X and Y experiment. Given the Stark shift is the third order polynominal of tone amplitude x, the P1 values for X and Y are ``` Px(x) = A cos(2 pi t_S (c1 x + c2 x**2 + c3 x**3 + f_err)) + offset Px(x) = A sin(2 pi t_S (c1 x + c2 x**2 + c3 x**3 + f_err)) + offset ``` where t_S is the fixed Stark delay and others are the fit parameters. Note that current fitter assumes that A is independent of x, however, since this is a variant of Ramsey experiment, the P1 amplitude actually depends on x. This is because qubit T2 may depend on the frequency, or at larger Stark amplitude the qubit heating may cause faster dephasing. In the new fitter, we directly extract the phase polynominal and perform fit on this synthesized data. Namely, ``` poly = unwrap(arctan2(Py, Px)) ~ 2 pi t_S (c1 x + c2 x**2 + c3 x**3 + f_err) ``` Because A is canceled out in this form, the fitting becomes robust to the amplitude dependence of the dephasing rate. Usually we cannot obtain the reasonable model for A(x). For example, the experiment data below shows the difference of fit on the poly data and raw Px, Py data (actually the parameters are fit on the poly data and Px, Py data are just visualized with the fitted parameters). ![image](https://github.com/Qiskit-Extensions/qiskit-experiments/assets/39517270/d14b8537-dcc9-489c-b277-fe7df7af8938) As you can see, the Px, Py data are damped quickly on positive amplitudes, and the envelope of the trigonometric function is asymmetric with respect to 0. In this situation the current fitter doesn't work well. In contrast, new phase fitter works nicely. --------- Co-authored-by: Will Shanks <[email protected]> Co-authored-by: Will Shanks <[email protected]>
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