## 3 Schrieffer-Wolff Rank-Reducing Similarity Transformations  
In this Section, we discuss the possibility of designing unitary transformation generated by the time-independent anti-Hermitian operator \(B\) (see Equation (7)). \[G=e^{B}He^{-B}\, \tag{23}\]  
that assures specific properties of the \(G\) operator and consequently simplifies qubit mapping of the evolution operator \(\Omega(t)\). In particular, we will prove the following Theorem:  
Theorem 3.1: _There exist conditions for anti-Hermitian \(B\)-operator that render the \(G\) operator in the following form: \(G=\widehat{\mathcal{P}}(G)+\widehat{\mathcal{P}}_{d}(G)+\widehat{\mathcal{P} }_{ie}(G)\). Moreover, the action of \(\widehat{\mathcal{P}}_{d}(G)+\widehat{\mathcal{P}}_{ie}(G)\) on the qubits corresponding to external spin-orbitals (assuming the ordering of spin-orbitals and qubits in accordance with Equation (8)) is local._  
Proof of Theorem 3.1.: To eliminate the \(\widehat{\mathcal{P}}_{eod}(G)\) we impose the condition for the \(B\) operator  
\[\widehat{\mathcal{P}}_{eod}(G)=\widehat{\mathcal{P}}_{eod}(e^{B}He^{-B})=0\, \tag{24}\]  
which can explicitly be expanded in terms of multi-commutator expansion using Hausdorff expansion  
\[\widehat{\mathcal{P}}_{eod}(H)-\widehat{\mathcal{P}}_{eod}([H,B])+\frac{1}{2} \widehat{\mathcal{P}}_{eod}([[H,B],B])-\ldots=0. \tag{25}\]  
Figure 2: A typical example of the external off-diagonal isoenergetic excitation operator \(E_{e(a)e(a)}^{\alpha}\). Solving the above equations in the Fock space requires some attention. One should realize that the dimensionality of Equations (25) is much bigger than the number of non-zero terms contributing to \(\widehat{\mathcal{P}}_{\text{e}od}(H)\) in Equation (25) and is equal to the number of all excitation-type operators spanning \(\widehat{\mathcal{P}}_{\text{e}od}\) space in Equation (13). The perturbative analysis of the solution of Equation (24) (see Appendix A) shows that at the first order of perturbation theory, the number of amplitudes defining \(B\) is precisely equal to the number of non-zero elements of \(\widehat{\mathcal{P}}_{\text{e}od}(H)\). However, higher orders of many-body perturbation theory generate higher many-body components in space \(\widehat{\mathcal{P}}_{\text{e}od}\). For this reason, in the general case, the \(B\) operator is expressed in terms of all operators in set (13) and satisfy the condition  
\[B=\widehat{\mathcal{P}}_{\text{e}od}(B)\, \tag{26}\]  
which guarantees that the number of equations and unknowns are equal. As will be discussed later, various approximate techniques can be used to approximate the \(B\) operator using a smaller number of variables. If the solution of Equation (24), labeled as \(B^{\star}\), can be found (or effectively approximated by limiting the rank of the multi-commutator expansion) on classical computers, then the \(G\) operator with the desired properties is given by  
\[G=\widehat{\mathcal{P}}(e^{B^{\star}}He^{-B^{\star}})+\widehat{\mathcal{P}}_{d }(e^{B^{\star}}He^{-B^{\star}})+\widehat{\mathcal{P}}_{ie}(e^{B^{\star}}He^{- B^{\star}}). \tag{27}\]  
The parts of \(G\) contributing to action on the qubits corresponding to external spin-orbitals are \(\widehat{\mathcal{P}}_{d}(e^{B^{\star}}He^{-B^{\star}})\) and \(\widehat{\mathcal{P}}_{ie}(e^{B^{\star}}He^{-B^{\star}})\). These two classes of elements are advantageous components of \(G\) when designing quantum circuits with low complexity. The diagonal part, according to Equation (21), obviously involves the particle number operators that are qubit-local. On the other hand, the encoding of general operators contributing to the \(\widehat{\mathcal{P}}_{ie}(e^{B^{\star}}He^{-B^{\star}})\) requires encoding a chain of \(E^{a}_{e(a)}/E^{a}_{e(a)}\) and their Hermitian conjugates. It turns out, however, that these operators involve gates only on the pairs of adjacent qubits \([Q\uparrow][Q\downarrow]\) in the representation given by scheme (8) (assuming a non-degenerate character of orbital energies as shown in Figure 1).