Adding gc sign conventions

[marekgluza]

I consolidate DoubleBracketIteration.__call__ to follow the convention that $H' = U^\dagger H U$ where $U=e^{-sW}$ with $W=[D,H]$ (or depending on mode an approximation, see eval_dbr_unitary). If $s>0$ then for $D = \Delta(H)$ the GWW DBR will give a $\sigma$-decrease, see https://arxiv.org/abs/2206.11772.

There is no clash because the dbr unitary is being exposed only now.

In call we will are working in the convention that $H' = U^\dagger H U$ where $U=e^{-sW}$ with $W=[D,H]$ or an approximation of that by a group commutator. That is handy because if we switch from the DBI in the Heisenberg picture for the Hamiltonian, we get that the transformation of the state is $|\psi'\rangle = U |\psi\rangle$ so that $\langle H\rangle_{\psi'} = \langle H' \rangle_\psi$ (i.e. when writing the unitary acting on the state dagger notation is avoided).
The group commutator must approximate $U=e^{-s[D,H]}$. This is achieved by setting $r = \sqrt{s}$ so that
$$V = e^{-irH}e^{irD}e^{irH}e^{-irD}$$
because
$$e^{-irH}De^{irH} = D+ir[D,H]+O(r^2)$$
so
$$V\approx e^{irD +i^2 r^2[D,H] + O(r^2) -irD} \approx U\ .$$
See the app in https://arxiv.org/abs/2206.11772 for a derivation.

Checklist:

• Reviewers confirm new code works as expected.
• Tests are passing.
• Coverage does not decrease.
• Documentation is updated.

[Edoardo-Pedicillo]

Hi @marekgluza looks good to me, please fix the tests and ask me the review again.

[codecov]

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[andrea-pasquale]

Thanks @marekgluza.
I only have a few minor comments.