## 4 QPE Formulations Based on the SW-RRST Representation of Many-Body Hamiltonians  
The main idea of the QPE is in the controlled execution of the powers of the \(\Omega(t)\) operator according to the progression  
\[\Omega(t)^{2^{0}}\rightarrow\Omega(t)^{2^{1}}\rightarrow\ldots\Omega(t)^{2^ {j}}\ldots\ldots\Omega(t)^{2^{m-1}} \tag{43}\]  
where \(m\) designates the number of ancilla qubits used to read the phase(s) of the unitary evolution operator (see Figure 3a). When a standard representation of the Hamiltonian \(H\) is used, the qubit encoding of the \(2^{j}\)-th power of \(\Omega(t)\) operator,  
\[\Omega(t)^{2^{j}}=(e^{-iH})^{2^{j}}\, \tag{44}\]  
utilizes all \(N\) \"physical\" qubits. When representation (7) is invoked, the same operator power can be expressed as  
\[\Omega(t)^{2^{j}}=(e^{-B}e^{-itG}e^{B})^{2^{j}}=e^{-B}(e^{-itG})^{2^{j}}e^{B}\, \tag{45}\]  
where \(B\) and \(G\) operators are given by Equations (7) or (23). The main difference between (44) and (45) (see Figure 3 insets (b) and (c)) is that the sequence \((e^{-itG})^{2^{j}}\) can be executed using localized qubit gates in the sense of earlier discussion of \(G\)-operator properties. The additional advantages stem from the fact that the external part of the \(G\) operator \(\widehat{\mathcal{P}}(G)\) involves a simpler form of the gates and large classes of operators that commute with the internal part of the \(G\) operator, \(\widetilde{\mathcal{P}}(G)\), which simplifies the form of the Trotter formula. We will explain it on the example of the \(G\) operator discussed in Corollary 1, where \(G\) is decomposed into internal \(\widetilde{\mathcal{P}}(G)\) and external \(\widehat{\mathcal{P}}(G)\) parts, and where the external part is expressed in terms of number operators \(\widehat{\mathcal{P}}(G)=f_{\mathcal{G}}(\{n_{p}\}_{p=1}^{N})\) since \(\widehat{\mathcal{P}}(G)\) only contains diagonal elements as a result of Corollary 1. Let us further decompose \(\widehat{\mathcal{P}}(G)\) into the part that mixes particle number operators for active and external spin-orbitals (\(\widehat{\mathcal{P}}_{M}(G)\)) and the part that is solely expressed in terms of the particle number operator corresponding to external spin-orbitals only (\(\widehat{\mathcal{P}}_{E}(G)\)). Therefore the following commutation relations holds  
\[[\widetilde{\mathcal{P}}(G),\widehat{\mathcal{P}}_{E}(G)]=[\widehat{\mathcal{ P}}_{M}(G),\widehat{\mathcal{P}}_{E}(G)]=0\, \tag{46}\]  
therefore  
\[e^{-itG} = e^{-it[\widetilde{\mathcal{P}}(G)+\widehat{\mathcal{P}}_{M}(G)]} e^{-it\widehat{\mathcal{P}}_{E}(G)} \tag{47}\] \[= e^{-it\widehat{\mathcal{P}}_{E}(G)}e^{-it[\widehat{\mathcal{P}}( G)+\widehat{\mathcal{P}}_{M}(G)]}\,\]  
where the \(e^{-it[\widetilde{\mathcal{P}}(G)+\widehat{\mathcal{P}}_{M}(G)]}\) term requires a Trotter formula to be implemented. At the same time the \(e^{-it\widehat{\mathcal{P}}_{E}(G)}\) terms as depending only on the particle number operators (or their products) can be calculated exactly, which is a consequence of the fact that all \(n_{p}\)'soperators defining \(\widehat{\mathcal{P}}_{E}(G)\) commute. For example, the JW qubit mapping of \(n_{p}\) and \(n_{p}n_{q}\) can be represented by a simple local circuits  
\[n_{p} \rightarrow \frac{1}{2}(1_{p}-\sigma_{p}^{z})\, \tag{48}\] \[n_{p}n_{q} \rightarrow \frac{1}{4}(1_{p}-\sigma_{p}^{z})\otimes(1_{q}-\sigma_{q}^{z})\, \tag{49}\]  
where \(1_{p}\) and \(\sigma_{p}^{z}\) are identity and Pauli Z matrices (gates) on \(p\)-th qubit.