  We derive recursive relations for the Schrieffer--Wolff (SW) transformation
applied to the half-filled Hubbard dimer. While the standard SW transformation
is set to block-diagonalize the transformed Hamiltonian solely at the first
order of perturbation, we infer from recursive relations two types of
modifications, variational or iterative, that approach, or even enforce for the
homogeneous case, the desired block-diagonalization at infinite order of
perturbation. The modified SW unitary transformations are then used to design
an test quantum algorithms adapted to the noisy and fault-tolerant era. This
work paves the way toward the design of alternative quantum algorithms for the
general Hubbard Hamiltonian.
