## 5 Approximations  
The potential of reducing hardware requirements discussed in the previous Section is inextricably connected to our ability to approximate solutions of non-terminating expansions (15) using their finite-rank commutator expansion (16). We envision this step to be entirely performed using conventional computers. This step is justified because the process of solving equations for finite-rank commutator expansions is associated with polynomial scaling.  
In analogy to the existing quantum algorithms, including a broad class of VQE and QPE formulations, classes of approximations are indispensable to translate the problem into the form of circuits for quantum computers or their classical emulators. For example, the broad class of Trotter-based approximations is needed for realizing unitary CC-driven variants of the VQE formalism and evolution operators in the QPE formalism. It does not come as a surprise that for the SW-RRST approximation, the situation is no different. The main factors that need to be taken into account when defining approximate SW-RRST formulations are as follows:  
1. **The excitation-rank of the many-body form of the transformed \(G\) Hamiltonian.** Our experience with Hilbert-space-type downfolding indicates that one- and two-body effective interactions can provide satisfactory results when active space is adequately defined [56].
2. **The rank of the many-body effects in the \(B\) operator.** The elementary perturbative analysis for the case discussed in Corollary 1 indicates that the 0-th order of \(B\) is equal to zero, the 1-st order contributed to one- and two-body terms, while the 2-nd order introduces higher-rank effects (see the analysis in Appendix A). It suggests that simple models based on the inclusion of one- and two-body effects in the \(B\) operator are justified.
3. **The working equations for \(B\)-amplitudes.** Due to their non-terminating nature, the algebraic form of Equations (24) or (36) has to be approximated due to their non-terminating nature. A numerically feasible way of introducing sufficiency conditions for \(B\)-operator amplitudes is to use finite commutator-rank expansion in Equations (24) or (36). The class of approximations termed SW-RRST(\(I\)) consist of retaining commutators up to the \(l\)-th rank. For formulation (24) we have  
\[\widehat{\mathcal{P}}_{\text{e}od}(H+\sum_{i=1}^{l}(-1)^{i}\frac{1}{i!}[\ldots[H, B],\ldots],B]_{i})=0. \tag{63}\]  
whereas for (36) one gets  
\[\widehat{\mathcal{P}}_{\text{e}d}(H+\sum_{i=1}^{l}(-1)^{i}\frac{1}{i!}[\ldots[H,B],\ldots],B]_{i})=0. \tag{64}\]  
While the linear SW-RRST(1) is the simplest approximation consistent with the low-order perturbative analysis, one may have to deal with the singular or nearly singular forms of the equation. Similar problems have been observed in the early studies of the CC theory [104]. Several techniques akin to almost-linear CC approximations are discussed in Refs. [105; 106; 107], including the auxiliary transformation discussed earlier, have the potential to offset singular behavior. Another factor that was shown to play a key role in the removal of singularities of linear CC equation is associated with the inclusion of non-linear terms, which is the main motivation for the development of higher-rank SW-RRST(I) approximations (\(l>1\)).  
Another important aspect of the proposed approximation is the choice of the orbital basis that should be driven by the targeted quantum system.