Lanczos method

[MatP1337]

This pull request add a complete implementation of Exact and efficient Lanczos method on a quantum computer for ground state energy estimation.

The quantum Lanczos method efficiently constructs a Krylov subspace by applying Chebyshev polynomials of the Hamiltonian to an initial state. It avoids the classical barrier of exponential cost in representing Krylov vectors.

Implements the following steps:
1. inner_lanczos: construct the circuit
2. lanczos_expvals: runs quantum Lanczos subroutine to obtain Chebyshev expectation values $\langle T_k(H)\rangle$.
3. build_S_H_from_Tk: builds overlap and Hamiltonian subspace matrices $(\mathbf{S}, \mathbf{H})$.
4. regularize_S_H: regularizes overlap matrix $\mathbf{S}$ and $\mathbf{H}$ by projecting onto the subspace with well conditioned eigenvalues.
5. lanczos_alg: utilizes all of the above to solve the generalized eigenvalue problem $\mathbf{H}\vec{v}=\epsilon\mathbf{S}\vec{v}$. Returns lowest eigenvalue $\epsilon_{\text{min}}$ as ground state energy estimate.

To-do:

• make compatible with jasp
  • Jasp prefix control now works in non-Jasp #314 fixes an occurring bug
• update documentation to functions
• add test(s)

[renezander90]

Auxiliary methods (inner_lanczos, lanczos_expvals) are Jasp compatible and jax versions of classical post processing functions are implemented. However, solving the generalized eigenvalue problems seems to be not implemented for the corresponding jax.numpy/jax.scipy functions (at least for version 0.6). Hence, executing full Lanczos method in Jasp is currently not supported.

Update: Lanczos implementation is now fully Jasp compatible using a custom implementation for solving the generalized eigenvalue problem

[renezander90]

We should also mention the numerical instability of the generalized eigenvalue problem, and the importance of regularization (a wrong result for the energy could be returned depending on the cutoff).

[renezander90]

We should could add some formulas for derivation of the generalized eigenvalue problem form expectation values <T_k(H)>.

[renezander90]

In principle, we could always use the jitted versions of the post processing functions, i.e., remove the check_for_tracing_mode distinction in the Lanczos_alg function. A minor drawback would be that for Jax.jit compatibility the shapes of the regularized matrices must be static. This is addressed by returning block matrices where the "true" regularized matrices are in the upper left block, identity in the lower right block; and zeros otherwise. This can be inconvenient to work with in static mode, but additionally returning the block size of the relevant block could be helpful.
Some further testing of the jitted versions would be required.