Improving integration of quantum_info module and qibojit

[renatomello]

Right now, even though the quantum_info module is compatible to all backends, it is not making use of any of the jit properties from numba. Therefore, numpy and numba performances are the same across the board. It would be good to profile the module and see where it is worth to invest time to code custom operations in qibojit.

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1. Parallelization of CliffordOperations through numba and cupy qibojit#161
   enhancement +1
   
2. Customizing Clifford operations for qibojit engines #1150
   enhancement quantum_info module +2
   
3. Optimizations for the qibo.quantum_info.basis.pauli_basis and vectorization function #1459
   documentation enhancement quantum_info module +3
   

[renatomello]

@AlejandroSopena @BrunoLiegiBastonLiegi

This is list of the functions that have priority in being optimized in qibojit:

Basis:

• comp_basis_to_pauli
• pauli_to_comp_basis

Random ensembles:

• random_statevector
• random_density_matrix
• random_unitary
• random_quantum_channel
• random_clifford

Superoperator transformations:

• vectorization
• unvectorization
• _reshuffliing
• to_pauli_liouville
• choi_to_kraus
• kraus_to_choi
• kraus_to_stinespring
• stinespring_to_kraus

[BrunoLiegiBastonLiegi]

@renatomello should we create a separate module in quantum_info, something like _utils.py and create the relative modules in qibojit for numba and cupy for this?
Then I thought we could delegate to the backend the loading of this utils through a method. For example in NumbaBackend:

@staticmethod
def _load_quantum_info_utils():
    from qibojit.quantum_info import _utils_cpu
    
    return _utils_cpu

and then the functions inside the quantum_info module will use the provided backend to load the utils and call them.

[renatomello]

@renatomello should we create a separate module in quantum_info, something like _utils.py and create the relative modules in qibojit for numba and cupy for this? Then I thought we could delegate to the backend the loading of this utils through a method. For example in NumbaBackend:

@staticmethod
def _load_quantum_info_utils():
    from qibojit.quantum_info import _utils_cpu
    
    return _utils_cpu

and then the functions inside the quantum_info module will use the provided backend to load the utils and call them.

Which routines would be there?

[BrunoLiegiBastonLiegi]

Which routines would be there?

All the ones we want to improve with numba and cupy

[renatomello]

Which routines would be there?

All the ones we want to improve with numba and cupy

Then they'd have to be subroutines of the functions we have, because the functions we want to improve are all user-facing and part of specific submodules

[BrunoLiegiBastonLiegi]

@AlejandroSopena @BrunoLiegiBastonLiegi

This is list of the functions that have priority in being optimized in qibojit:

Basis:

* [ ]  `comp_basis_to_pauli`

* [ ]  `pauli_to_comp_basis`

Random ensembles:

* [ ]  `random_statevector`

* [ ]  `random_density_matrix`

* [ ]  `random_unitary`

* [ ]  `random_quantum_channel`

* [ ]  `random_clifford`

Superoperator transformations:

* [ ]  `vectorization`

* [ ]  `unvectorization`

* [ ]  `_reshuffliing`

* [ ]  `to_pauli_liouville`

* [ ]  `choi_to_kraus`

* [ ]  `kraus_to_choi`

* [ ]  `kraus_to_stinespring`

* [ ]  `stinespring_to_kraus`

#1459 is trying to address the first two boxes (basis section)