Drag fit instability

[eggerdj]

Summary

This PR solves #425

Details and comments

The fit function for the drag experiment is

y_i(x) = amp * cos(2 * pi * freq_i * x - 2 * pi * beta) + base   # i in [0,1,2] i.e. three curves

This will yield the amp, freq_0, freq_1, freq_2, beta, base which best fit the y-data.
The DragCal is aimed at determining the beta at which all three curves are minimum, i.e. y_i=0.
This implies

- amp * cos(...) = base          Eq. (1)

The fit will typically return base=0.5 and amp = \pm 0.5. There is therefore an ambiguity in the sign of the amplitude which can be compensated for by the cosine term to satisfy Eq. (1).

• If amp > 0 we need cos(...) = -1 which will occur for 2 * pi * freq_i * x - 2 * pi * beta = pi mod 2 pi.
• However, if amp < 0 we need cos(...) = 1 which will occur for 2 * pi * freq_i * x - 2 * pi * beta = 0 mod 2 pi.

These two scenarios therefore create an instability in beta since the two cases (positive and negative amp) can be accommodated by different beta values. To overcome this we restrict the value of the amplitude to negative values, i.e. the default bounds are now (-2 * y_max, 0) instead of (-2 * y_max, 2 * y_max). Note that users can always override these bounds.

Note 1: I also removed the Kernel test since this does not have the concept of 0 and 1 state meaning that the signal is sometimes flipped. I realize that for the DragCal we need to know where the |0> state is.

Note 2: I increased the test value from 0.005 to 0.0051 since now it fails at 0.005. This sensitivity is not great but the test is also not super since one should rather increase the range of beta values rather than trying to fit cosines to flat curves.

[wshanks]

I am approving because this should take care of the immediate issue.

There are a few other changes I would recommend:

1. The analysis does not enforce that the minimum beta value is chosen. It is possible if one sweeps over a wide enough range that there is more than one zero. It would be best to look at the frequencies fitted and make sure the smallest beta value was found.
2. The frequencies should scale with the number of iterations. So a single fit frequency could be used for the three curves.
3. There has been some discussion of not using bounds for the fit parameters. The amp bound could be removed by instead checking the sign of the fitted amp and choosing the right zero of the oscillation as the beta value, or by making the fit function use abs(amp).

[eggerdj]

Thanks Will. I'll open an enhancement issue and close the bug issue.