## 2 Rank-Reducing Unitary Similarity Transformations of Many-Body Hamiltonians  
The dynamics of the quantum system are given by the evolution operator \(\Omega(t)\)  
\[\Omega(t)=e^{-itH} \tag{1}\]  
where we assume that the Hamiltonian \(H\) is time-independent and takes the following second-quantized form in the basis of \(N\) spin-orbitals  
\[H=\sum_{p,q=1}^{N}h_{q}^{p}a_{p}^{\dagger}a_{q}+\frac{1}{4}\sum_{p,q,r,s=1}^{ N}v_{rs}^{pq}a_{p}^{\dagger}a_{q}^{\dagger}a_{s}a_{r}\;, \tag{2}\]  
where \(p,q,r,s\) are spin-orbital indices and \(h_{q}^{p}\) and \(v_{rs}^{pq}\) are one- and two-electron (anti-symmetrized) integrals defining \(H_{1}\) and \(H_{2}\) operators, respectively. The \(a_{p}^{\dagger}\) (\(a_{p}\)) operator corresponds to a creation (annihilation) operator for the electron in \(p\)-th spin-orbital. In quantum computing applications, especially in quantum phase estimations, various representations of the electronic Hamiltonian (induced, for example, by the unitary transformations) can be used. This is a consequence of the fact that the spectrum of the Hamiltonian remains unchanged upon these transformations. Let us denote a general unitary transformation \(U\) as  
\[U=e^{A(t)}\, \tag{3}\]  
where the operator \(A(t)\) is anti-Hermitian  
\[A(t)^{\dagger}=-A(t)\, \tag{4}\]  
and similarity-transformed Hamiltonian \(\tilde{H}(t)\) is defined as  
\[\tilde{H}(t)=e^{-A(t)}H(t)e^{A(t)}. \tag{5}\]  
In analogy to the \(H\) operator, the similarity-transformed Hamiltonian is also Hermitian. In consequence, probing the phase with the \(\tilde{H}\) and corresponding time evolution operator \(\tilde{\Omega}(t)\)  
\[\tilde{\Omega}(t)=e^{-itB(t)}=e^{-A(t)}\Omega(t)e^{A(t)} \tag{6}\]  
should detect the same values of energy/phase (subject to the various choices of initial state). A typical illustration of the above techniques is the interaction and Heisenberg pictures widely used in quantum mechanics, which have recently been explored in the context of quantum computing simulations. The interaction-picture-based approach has recently been studied in the context of quantum computing [94; 95; 96]. In this paper, we will pursue a slightly different goal associated with the design of _time-independent_ unitary transformation generated by the time-independent anti-Hermitian operator \(B\) (\(B^{\dagger}=-B\)) such that  
\[H=e^{-B}Ge^{B}\, \tag{7}\]  
where the properties of the \(G\) operator and the form of the \(B\) operator assuring these properties will be discussed later. For the sake of the following discussion, let us introduce the partitioning of the orbitals (spin-orbitals) into active (with corresponding first \(2(n-k)\) qubits) and external (with qubits enumerated as \(2(n-k)+1,\dots,2n\)) as shown in Figure 1. Additionally, we will assume that all spin-orbitals are arranged in a way that spin-up (\(\uparrow\)) and spin-down (\(\downarrow\)) spin-orbitals occupying the same orbitals \(P\) (isoenergetic spin-orbitals) are neighboring as shown in the following scheme. \[\dots[Q\uparrow][Q\downarrow]\dots[P\uparrow][P\downarrow]\dots \tag{8}\]  
If spin-orbitals \(p\) and \(q\) are isoenergetic we will denote it by \(q=e(p)\) (or \(p=e(q)\)). Figure 1: The orbital (spin–orbital) domain is partitioned into \(n-k\) active and \(k\) external orbitals (\(n\) stands for the total number of orbitals). The active spin-orbitals are denoted as \(\alpha\), \(\beta\),..., the external spin–orbitals as \(a\), \(b\),..., and generic spin–orbitals as \(p\), \(q\),.... In general, the active spin–orbitals do not have to be defined as the “lowest” lying spin–orbitals using some energy-related criteria. At this moment, the nature of active/external spin–orbitals remains unspecified. The total number of spin–orbitals \(N\) is defined as \(N=2n\). For simplicity, in this paper, we assume that the orbital energies are non-degenerate.