Feat - Pegasos Quantum Kernel Alignment

[JohnKerf]

Summary

• PR related to Add support for trainable quantum kernels to Pegasos Quantum Support Vector Classifier #946.
• Extends the PegasosQSVC algorithm to perform quantum kernel alignment within the optimization loop.
• Based on the logic introduced in https://ieeexplore.ieee.org/document/10313634 with accompanying code https://zenodo.org/records/7804477.
• Adapts the TrainableKernel classes to allow for evaluating trainable kernels with different left and right parameters.

Details and comments

• The majority of the PegasosQKA code follows from https://zenodo.org/records/7804477 but has been stripped back to make the implementation more streamlined.

• Decided not to include the self._blasphemy logic (computing kernel without historical thetas) as this is not how the algorithm is intended to be ran.

• Decided at this time not to include a callback however this could be a possible future feature.

• Precomputed kernel functionality is not needed for QKA since by design we change the kernel throught the algorithm.

• Decided warm starting and calibration/recalibration is not needed. It seems calibration was not used in the paper and adds a lot of extra bloat to the algorithm so I thought it was best to not include it (learning_rate and perturbations are passed as fixed variables).

• To be able to cleanly inherit PegasosQSVC I extracted the step update logic into its own function _update_step and also the kernel evaulation logic into _evaluate_kernel. This meant we can cleanly overide these in the PegasosQKA class and reuse the bulk of the QSVC algorithm.

• Previously the trainable kernels did not have the option to set seperate paramters for the left and right feature maps. The paper logic introduced a new pseudo kernel but to avoid adding this I have introduced the ability to run and evaluate the trainable kernels with the seperate left and right parameters.

AI Disclosure - *ChatGPT was used in this work primarily to help with producing tests and also helped me better understand the PegasosQSVC/PegasosQKA algorithms. The main body of code already existed from the paper source code and all implementation/checks were done by myself. I fully understand the code I have produced but I will note that during the process ChatGPT did help verify/clarify some of the logic I was imlpementing. *