Implement optimization strategies for functions of orthogonal matrices

[ecpeterson]

Implement generic versions of some of the algorithms described here: https://arxiv.org/abs/physics/9806030 .

[jmbr]

I have used this stuff for a side-project recently via pymanopt. If there's a (business) case for implementing this functionality in MAGICL but you're currently too busy, then I'd love to take care of it.

[ecpeterson]

Definitely! The business case is the one described in VII.A here: https://arxiv.org/pdf/1904.10541.pdf . The "searching" compiler templates in src/compilers/approx.lisp currently use a high-dimensional Nelder-Mead instance to optimize a function over the space of single qubit operators, which is simultaneously slow and unreliable. After conjugation by the "magic operator" (i.e., +e-basis+), it's equivalent to optimize the (smooth) function over the space of orthogonal operators, which I'd hope would be much speedier.

I tried implementing this some months ago (wow! really, in February?) and got myself into trouble with poor numerics. I'd be very glad to have it implemented by someone else.