Fock-Space Schrieffer–Wolff Transformation: Classically-Assisted Rank-Reduced Quantum Phase Estimation Algorithm

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.

Excited-state downfolding using ground-state formalisms

“…Mathematically rigorous formulations for reducing the dimensionality/cost of quantum formulations are necessary for expanding the envelope of system sizes tractable with current and near-term quantum simulators and hardware. There have been a variety of approximations and techniques in recent years for reducing the dimensionality and the complexity of quantum calculations [26][27][28][29][30][31][32][33][34]. One of the most promising formalisms is the downfolding technique based on the double unitary coupled cluster (DUCC) ansatz [25,[35][36][37][38][39][40], which constructs effective (or downfolded) Hamiltonians in a small-dimensionality sub-space of the entire Hilbert space, which is commonly defined as an active space.…”