Compiling xxz

[marekgluza]

This PR delivers the current code which compiles into CNOT+R different DBI strategies:

1. Start by running https://github.com/qiboteam/boostvqe/blob/compiling_XXZ/notebooks/running_boosting.ipynb

2. You can explore the different settings:

   • loading of different vqe results based on stating the number of qubits and layers etc
   • target epoch of the VQE circuit that will be boosted
   • how many steps will be evaluated
   • you can use different initializations which use different parametrizations of $D=H(B,J=0)$ or $D=H(B,J)$
   • number of gradient epochs
   • the notebook runs on verbose and visual so it will provide plots and outputs about the performance

3. The goal of this PR is to have a version of the code which will be evaluated on the cluster

   • the gradient descent routines give certain improvements but can be still optimized (how many evals one uses for s and lr optimization; how many gd steps actually help; from which profile to start from https://github.com/qiboteam/boostvqe/blob/53ec0131fe7eb1ef0a489794f6c6048d4b6b3db6/src/boostvqe/utils.py#L437C1-L450C1

4. This PR doesn't deliver systematic plots and storing - this remains to be done in the load_vqe.py conventions

Please review with priority - we need to run using this stuff asap so that we have results that we can discuss at TII.

Thanks @Sam-XiaoyueLi for helping out!

Also, thanks @khanhuyengiang @jeongrak-son and @shangtai for contributions

[khanhuyengiang]

Hi @marekgluza and @jeongrak-son, I have pushed the concise compiling code, see compiling_XXZ.py and compiling_XXZ_test.py. Please check and let me know if there is any issue. Thanks!

[marekgluza]

For future reference moving here our description of a previous feature:

Following Vatan and Williams https://arxiv.org/pdf/quant-ph/0308006

the 2 qubit Heisenberg gate defined in their notation as $N$ can be implemented with the circuit

This is a simplification from what Kay was trying to use and found a bug.

• [x] Please find the error in the notebook in notebooks/Compiling XXZ.ipynb for 2 qubits XXZ (Solution from Jeongrak: For 2q there is PBC.)

• [x] demonstrate that for whatever delta parameter of hamiltonians.XXZ it works by fixing appropriate angles Denote by $H^{(a)} = XaX{a+1}+ YaY{a+1} + \delta ZaZ{a+1}$ and so $H_{XXZ} = \suma H^{(a)}$. Show that we have $e^{-it H^{(a)}} = e^{-it H{XXZ, Qibo}}$ where the LHS is decomposed into CNOT and R.

• [x] demonstrate that it works for whatever time $t$

• [x] generalize the code to make a CNOT+R circuit which implements 1 and then M steps of the Trotter-Suzuki decomposition for $L$ qubits $$e^{-it H_{XXZ, Qibo}}=( \prod_a e^{-it/M H^{(a)}})^M$$

• [x] please demonstrate the polynomial scaling of the error $O(t^2/M)$ @khanhuyengiang please ask Jeongrak

• [ ] compare with a symbolic Hamiltonian .circuit

[marekgluza]

Closing for clarity: we will merge #52 into the master which will be easier to review