Recursive relations and quantum eigensolver algorithms within modified Schrieffer--Wolff transformations for the Hubbard dimer

TL;DR

This paper develops recursive relations for the Schrieffer--Wolff transformation in the Hubbard dimer, introducing modifications that enable higher-order block-diagonalization and informing the design of quantum algorithms for Hubbard models.

Contribution

It introduces recursive relations and modifications to the Schrieffer--Wolff transformation, facilitating improved quantum algorithms for the Hubbard Hamiltonian.

Findings

Recursive relations for the SW transformation derived.

Modified SW transformations approach infinite-order block-diagonalization.

Potential quantum algorithms for Hubbard models are proposed.

Abstract

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.