###### Abstract  
We present an extension of many-body downfolding methods to reduce the resources required in the quantum phase estimation (QPE) algorithm. In this paper, we focus on the Schrieffer-Wolff (SW) transformation of the electronic Hamiltonians for molecular systems that provides significant simplifications of quantum circuits for simulations of quantum dynamics. We demonstrate that by employing Fock-space variants of the SW transformation (or rank-reducing similarity transformations (RRST)) one can significantly increase the locality of the qubit-mapped similarity-transformed Hamiltonians. The practical utilization of the SW-RRST formalism is associated with a series of approximations discussed in the manuscript. In particular, amplitudes that define RRST can be evaluated using conventional computers and then encoded on quantum computers. The SW-RRST QPE quantum algorithms can also be viewed as an extension of the standard state-specific coupled-cluster downfolding methods to provide a robust alternative to the traditional QPE algorithms to identify the ground and excited states for systems with various numbers of electrons using the same Fock-space representations of the downfolded Hamiltonian. The RRST formalism serves as a design principle for developing new classes of approximate schemes that reduce the complexity of quantum circuits. quantum computing; practical quantum algorithms; quantum chemistry Article  
Fock-Space Schrieffer-Wolff Transformation: Classically-Assisted Rank-Reduced Quantum Phase Estimation Algorithm  
20231353991300539130  
Hermitian CC downfolding procedures were also discussed and tested in the context of quantum dynamics and excited-state simulations [52; 53; 55]. Due to the state-specificity of the downfolding procedures, the latter attempts require detailed knowledge of the external Fermionic degrees of freedom for excited states and the construction of separate effective Hamiltonians for each excited state. Although extraction of excited-state external correlation effects is possible for some classes of excited states, which can be captured by approximate Equation-of-Motion CC (EOMCC) methods [86; 87; 88], the generalization of this formalism to general-type excited states may be numerically challenging. A part of the problem is also associated with translating the EOMCC Ansatz defined by commuting operators into a language of unitary CC expansions involving non-commuting operators. Instead of following this strategy, in this paper, we discuss the class of Fock-space Schrieffer-Wolff (SW) transformation-inspired downfolding procedures designed to simplify the many-body form of the Hamiltonian. In analogy to the standard DUCC-based techniques, the SW-transformation-based (or rank-reducing similarity transformation (RRST)) formulations utilize the partitioning of one-electron states (spin-orbitals) into active and external spin-orbitals. Although the RRST cannot eliminate all components of the Hamiltonian that involve creation/annihilation operators carrying the external spin-orbital indices (the so-called external component of the Hamiltonian), the RRST is designed in a way that leads to a simple form that involves only local actions of qubits in corresponding quantum algorithms such as QPE. In the context of QPE methodology, this form not only makes qubit mappings simpler but can enable more efficient utilization of the Trotter formulas. In contrast to the standard DUCC downfolding, the proposed approach and related approximations eliminate its state-specific character and provide a description of multiple electronic states corresponding to the ground and excited states at least well approximated by the set of active orbitals. An exciting feature of the discussed framework is its universal character (in the sense of Fock space) in describing many-body systems with various numbers of electrons (particles), where the number of particles is specified when the action of the Hamiltonian (or the corresponding quantum evolution operator) on specific states takes place.