Transpiling double bracket iterations into hamiltonian simulation

[marekgluza]

This is a draft (code works ok and preparing for review)

This PR delivers a workeable structure for further development.
The point is to:

• show you the idea of the DBI transpiling
• ask you for your opinion for the design structure
• if you agree then to ask one of you to take it over and combine this with Circuit gate counting techniques
• the next 2 weeks I will not be able to work on this so let's make plans on matrix how to move ahead with gate counting

More specifically

• there are 2 new files in /dbi
• the goal was to propose a workflow which is capable of explaininig how the DBI transpiling into circuits must work (see overleaf for the revision of my paper)
• the idea is that there is now a class EvolutionOracle that can be working underneath with Circuit or numpy. Numpy we have in double_bracket.py so it's backwards compatible and Circuit e.g. for counting gates
• then there is the group commutator iteration file which is showing how to run the oracles. For the transpiling the key idea is the class FrameShiftedEvolutionOracle which recursively implements
  $$e^{it H_k} = e^{itU^\dagger H_0 U} = U^\dagger e^{itH_0} U$$
  which is what I refer to as frame shifting
• using that we have the evolution oracle of the iterated input hamiltonian so we know how to relate the group commutator step to
  $$V = e^{is D_k}e^{isH_k}e^{-is D_k}e^{-isH_k}$$
  the $H_0$ evolution
  $$V = e^{is D_k}U^\dagger e^{itH_0} Ue^{-is D_k}U^\dagger e^{-itH_0} U$$

Checklist:

• Reviewers confirm new code structure is reasonable.

Checklist:

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

[Edoardo-Pedicillo]

Here dbi_object is missing as input of dGamma_diDiagonal. About the naming, follow Python convention (https://peps.python.org/pep-0008/#naming-conventions), i.e. function names should be lowercase.

[marekgluza]

@wrightjandrew tomorrow Andrea will join us until 9:30 AM EU and we can ask more about your PR (your branch was merged into this one and this is something you might need to update in your PR cost function)

[marekgluza]

@Edoardo-Pedicillo @MatteoRobbiati @andrea-pasquale we realized that this revert commit
e2232c0
unfortunately undid in #1269 the #1287 PR of Edoardo which outputs the the unitary of DBI - sorry!

This means that the master is affected to the pre #1287 state.

The suggested fix is to merge this PR: the branch dbi_fix was merged into it this preserved the git blame history etc https://github.com/qiboteam/qibo/blame/DBI-transpiling-into-Hamiltonian-Simulation/src/qibo/models/dbi/double_bracket.py so the contribution is intact

[andrea-pasquale]

@Edoardo-Pedicillo @MatteoRobbiati @andrea-pasquale we realized that this revert commit e2232c0 unfortunately undid in #1269 the #1287 PR of Edoardo which outputs the the unitary of DBI - sorry!

This means that the master is affected to the pre #1287 state.

The suggested fix is to merge this PR: the branch dbi_fix was merged into it this preserved the git blame history etc https://github.com/qiboteam/qibo/blame/DBI-transpiling-into-Hamiltonian-Simulation/src/qibo/models/dbi/double_bracket.py so the contribution is intact

I see. I think that the easiest solution is just to open a small PR pointing to main that re-implements #1287.
Can you guys take care of it?
For the future, I know that tests were passing but I would let just coredevs merge PRs in master.

[marekgluza]

Yep, this was some 'auto merge' workflow and coredevs should merge in the future.

We will prepare the PR.