Minimize Energy fluctuation

[MatteoRobbiati]

No description provided.

[wrightjandrew]

I've already derived an expansion for the fluctuation that gives a third-order polynomial:
$$ \Xi_k^2 (\mu) &= \langle \mu | H_k^2| \mu \rangle - \langle \mu | H_k| \mu \rangle^2 \
&\approx \langle \mu | (\Gamma_0 + s\Gamma_1 + \frac{s^2}{2} \Gamma_2 + \frac{s^3}{6} \Gamma_3)^2| \mu \rangle -\langle \mu | \Gamma_0 + s\Gamma_1 + \frac{s^2}{2} \Gamma_2 + \frac{s^3}{6} \Gamma_3| \mu \rangle^2\
&\approx \text{Var}(\Gamma_0) +2s\text{Cov}(\Gamma_0,\Gamma_1)+s^2(\text{Var}\left(\Gamma_1)+\text{Cov}(\Gamma_0,\Gamma_2)\right)+\frac{s^3}{3}\left(\text{Cov}(\Gamma_0,\Gamma_3)+3\text{Cov}(\Gamma_1,\Gamma_2)\right) $$
https://www.overleaf.com/read/fcqswmyqcskf# (see part 4)

Finding the roots of its derivative gives an estimation of the optimal rotation step.
I've got some formulas for applying gradient descent for finding D but they also depend on how we parametrize this operator. This is also applicable for the full DBI and I think it may be good to discuss how the general case should be implemented first

[marekgluza]

Almost merged into master qiboteam/qibo#1269
code in use so closing, Thanks!