Coupled Cluster Downfolding Theory: towards universal many-body algorithms for dimensionality reduction of composite quantum systems in chemistry and materials science

Abstract

The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems commonly encountered in materials science and chemistry. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.

Excited-state downfolding using ground-state formalisms

“…Once again, the µ notation is used because the DUCC approach can describe excited states with a proper reference determinant choice despite starting as a ground-state formalism. The DUCC formalism is described in greater detail in [25,35,40], but the foundation of the downfolding approach is that the energy E µ can be obtained by diagonalizing an effective Hamiltonian Heff(DUCC) ext in the corresponding active space (defined by projection operator P + Q int , where P and Q int are projection operators onto the reference function and electron-promoted determinants in the active space, respectively)…”

“…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. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space.…”

Machine Learning Aided Dimensionality Reduction towards a Resource Efficient Projective Quantum Eigensolver

“…We will refer our scheme as ML aided PQE(ML-PQE). In this context we must mention that certain other downfolding techniques have been recently developed to solve classical coupled cluster theory using some sub-system embedding subalgebra (SES) techniques [37][38][39][40][41] and later extended to quantum computing framework as well [42][43][44][45][46].…”

Iterative quantum phase estimation with variationally prepared reference state

“…In general, a highly parametrized ansätz leads to higher expressibility, but also suffers from high circuit depth and trainability [29]. Other notable works in this direction include: Qubit Coupled Cluster (QCC) [30], Orbital Optimized Unitary Coupled Cluster (OO‐UCC) [31, 32], Double Unitary Coupled Cluster (DUCC) [33, 34], Adaptive Derivative‐Assembled Pseudo‐Trotter Variational Quantum Eigensolver (ADAPT‐VQE) [35], MR‐UCCpGSD [36] and the Projective Quantum Eigensolver (PQE) [37] approaches.…”