Add an Hybrid HHL notebook

[TomerGoldfriend]

In this issue, we will create an implementation of a hybrid HHL algorithm, following the ideas that appeared in Ref. [1]. The algorithm is based on the basic HHL algorithm for solving a set of linear equation, where the eigenvalue inversion part is relaxed by feeding-forward eigenvalue approximation from a QPE routine. The algorithm consists of three parts: (1) QPE, (2) classical signal post-prosseing, and (3) matrix inversion (HHL) with known eigenvalues.

This tutorial should follow the structure of the Deutsch Jozsa algorithm implementation, and take into account the technical comments outlined below.

To complete this issue, follow these steps:

1. Create a new jupyter notebook (.ipynb file). Use any jupyter editor (e.g. jupyter lab, google colab, etc).
2. Use Classiq's SDK to create a simple implementation of the hybrid HHL approach, and showcase the results (see technical comments below). If you have any implementation questions or challenges, the Classiq team will assist you, either on Github or in our slack community.
3. Create a short mathematical explanation of the work. Jupyter notebooks support markdown cells, which can contain LaTeX.
4. Make sure the notebook looks well, does not have any typos / mistakes, and is running properly.
5. Add the notebook to a new directory classiq-library/community/advanced_examples/hybrid_hhl/.
6. Follow the contribution guidelines to open a pull request.

Technical comments:

• The showcase should be for a generic random matrix, or an applicative use-case, and not for a matrix whose eigenvalues have an exact binary representation (as was done in Ref. [1]).
• Consider the following (optional):
  • Demonstrating for a small matrix and use an exact Hamiltonian evolution (using the built-in unitary function) instead of an approximated one (e.g., Suzuki-Trorrer etc.) to highlight the effect of the hybrid approach.
  • Reducing the QPE accuracy in the third HHL step, with respect to the accuracy in the first QPE step (as discussed in Ref[2]).
  • Exploring different optimization scenarios for the Synthesis.
  • Comparing the results to the textbook HHL approach.

If you have any questions or comments, you can ask them here in the issue, or in our slack community, and the Classiq team will be happy to assist.

Happy quantum coding!

References

[1]: Hybrid quantum linear equation algorithm and its experimental test on IBM Quantum Experience
[2]: Hybrid HHL with Dynamic Quantum Circuits on Real Hardware