Implement Mixed Magnitude Approximation

[Acciaccatura]

hi! i tried to attempt a solution for mixed magnitude approximation (https://arxiv.org/pdf/2203.10064) for #1759 - though I have a few issues that hopefully the Qualtran team can help me with? (forgive me, I don't have much QC experience!)

1. I produce an X under- and over-approximation by applying a rotation on the result of the mixed diagonal under- and over-approximations:

$$ \begin{align}\hat{X}_{under} = H\hat{Z}_{under}H^\dagger &\hat{X}_{over} = H\hat{Z}_{over}H^\dagger\end{align} $$

but a small percentage of the time (~0.2%) this produces two under-rotations - it seems the $H$ rotations add some error from the complex component of $Z_{1, 0}$. is this tolerable? if not, should I be trying to create my own ApproxProblem subclass for this?

2. I am unsure how to calculate the resulting diamond norm for probabilistic channels. is it simply $p * \Vert(\hat{U}_{under} - U)\Vert_◇ + (1 - p)\Vert(\hat{U}_{over} - U)\Vert_◇$? because with diamond norms calculated by UnitaryChannel.diamond_norm_distance_to_unitary like this, this results in a higher error than the solution says in Proposition 3.22 (my error is $> 3\epsilon$), despite my approximations seeming to satisfy the constraints of both Propositions 3.21 and 3.22.

thanks for taking a look!