Logarithmic mapper for BosonicOp

[ftroisi]

What should we add?

A Logarithmic mapper for the BosonicOp class

Purpose

Currently, Qiskit Nature offers many mappers for the FermionicOp class, but only the BosonicLinearMapper for the BosonicOp class.
As the name suggests, the number of required qubits to represent any bosonic operator scales linearly with the number of modes and Fock states used to represent each mode.
This strongly limits the possible applications of BosonicOp class, especially when dealing with mixed boson-fermion systems (see #795, #1188).

How

We need to add a BosonicLogarithmicMapper class, which inherits from BosonicMapper. This class will implement the mapping in Ref [1], in particular equation 7 (which defines how to map the creation operator). We will need to map the annihilation operator as well.
The formulas can also be taken from the paper in Ref [2], equations (34, 35), and from Ref [3] equation (28, 29).

Note that this paper is published only in the Arxiv, but they are already cited and they all share the same results. Therefore, since they are independent, we can safely that their claims and formulas are correct.

References

• [1] Huang et al., Qubitization of Bosons, https://arxiv.org/abs/2105.12563
• [2] Bo Peng et al., Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Construction, and Error Analysis, Arxiv https://doi.org/10.48550/arXiv.2307.06580
• [3] Veis et al., Quantum chemistry beyond Born-Oppenheimer approximation on a quantum computer: a simulated phase estimation study, https://doi.org/10.48550/arXiv.1507.03271

[ftroisi]

@Anthony-Gandon I'm tagging you here after today's discussion (for some reason I couldn't find you in the assignees)

@mrossinek I'm tagging you as a maintainer to keep you in the loop. We discussed with Ivano that this is becoming a priority. I'm happy to work on the implementation

[woodsp-ibm]

for some reason I couldn't find you in the assignees

In general it will take someone to make a comment in the issue before they will appear in the assignees dropdown