[Feature] Add quantum information tools

[inafergra]

This PR follows the work of this old qadence PR. It adds a subpackage qinfo_tools containing different quantum information-related functionalities. The three files of the subpackage are:

• qadence_libs/qinfo_tools/qfi.py contains code related to the calculation of the Quantum Fisher Information (QFI) matrix. The function get_quantum_fisher() calculates the QFI matrix of a QuantumCircuit in an exact way, while the get_quantum_fisher_spsa() can be used to iteratively construct an approximation of the QFI matrix through the SPSA algorithm.
• qadence_libs/qinfo_tools/spsa.pycontains the code to calculate the SPSA gradients. Only the second order gradient is used in the QNG-SPSA optimizer but the first order gradient is also implemented for completeness.
• The main contribution is the qadence_libs/qinfo_tools/qng.py file implementing the Quantum Natural Gradient (QNG) and the QNG-SPSA optimizers. Both optimizers use the QFI matrix in the parameter update rule to better navigate the gradient descent. The QNG uses the exact QFI matrix, which takes $\mathcal{O}(n^2)$ circuit evaluations per iteration ($n$ being the number of variational params in the circuit), making the QNG optimizer very slow. The QNG-SPSA uses the SPSA approximation of the QFI matrix (https://arxiv.org/pdf/2103.09232.pdf) resulting a much more efficient optimizer: it only takes 6 circuit evaluations per iteration to calculate the approximation of the QFI matrix (constant overhead $\mathcal{O}(1)$ ).

A new section in the docs has also been created for quantum information related topics, with a file explaining how to use the QNG and QNG-SPSA optimizers for an easy QCL problem. Since there was yet no files in the documentation, this PR also adds a docs/docsutils.py file with a fig_to_html() function to print images in the docs (same as in the Qadence repo) and a docs/environment.yml for the python environment (also same as in Qadence).

[RolandMacDoland]

Hey @inafergra thank you for this. Is this ready for review ? In case, you should un-draft it.

[inafergra]

Hey @inafergra thank you for this. Is this ready for review ? In case, you should un-draft it.

It is ready for review now

[gvelikova]

Thanks Ignacio! I think you should add the qinfo_tools imports to the init.py of the package in the same way as the constructors. I am testing it with some PDEs and will try to add a proper review soon.

[RolandMacDoland]

Thanks @inafergra. Is this reviewable ? Another quick question: would it make sense to move SPSA to PyQ as another differentiation method ?

[dominikandreasseitz]

Thanks @inafergra. Is this reviewable ? Another quick question: would it make sense to move SPSA to PyQ as another differentiation method ?

@Roland-djee we can add spsa in pyq but the implementation in this MR is highly geared towards ignacios usecase, it for example works only for overlap models which do not exist atm in pyq

[inafergra]

@Roland-djee I still need to add some tests related to the indexing order of the vparams. I'll mark this PR as a draft until it's done. Regarding the SPSA approximation, I agree with Dominik, it can make sense to place it in PyQ in the future but that will require generalizing the module quite a bit.

[inafergra]

The PR is ready for review now. I have tried to implement the changes as discussed in the comments and in our meetings, mainly:

• The QFI functions now take a dictionary with the variational parameters names and tensors, to avoid dependence on the ordering of the parameters (before they were passed as a Sequence).
• However, inside the QNG optimizer the parameters still need to be passed as a Sequence (this is the expected behaviour of the torch.optimizer base class). I assemble a dictionary inside the QNG optimizer to pass to the QFI functions and I make sure the ordering is correct in the tests.
• In the parameter update we now solve the least square problems instead of computing the inverse, as Joao pointed out this is way faster and more stable.
• I added a custom type for the different QFI approximations
• Added more tests, particularly for the symmetric matrix square root

[RolandMacDoland]

Thanks @inafergra. Looking good to me. Will approve as soon as comments are addressed.

[inafergra]

Thank you for the comments @Roland-djee, all addressed. Ready to merge

[RolandMacDoland]

Thanks @inafergra looking very nice.

[inafergra]

Thanks @Roland-djee, will merge now!