Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms

Abstract

In this paper we outline the extension of recently introduced the sub-system embedding subalgebras coupled cluster (SES-CC) formalism to the unitary CC formalism. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC formalism (DUCC) the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects -used to define the effective Hamiltonian -from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of complete active space representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction (FCI) type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing.

Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation

“…The sequential approach and NIVO approximations are general, and can be applied to improve the efficiency of other unitary nonperturbative methods (e.g., unitary coupled cluster theory) and downfolding schemes for classical-quantum hybrid algorithms. 46…”

“…The latter property is an important advantage in new applications of unitary methods to quantum computing, both in quantum algorithms [34][35][36][37][38][39][40][41][42][43][44][45] and downfolding approaches aimed at reducing the number of orbitals in quantum computations. 46 One of the main obstacles in the formulation of both single-and multi-reference unitary coupled cluster theories is that they lead to nonterminating equations. The central quantity evaluated in these approaches is the similarity transformed Hamiltonian (H) defined aŝ…”

NWChem: Past, present, and future

“…NWChem, with its computational infrastructure to characterize second-quantized forms of electronic Hamiltonians in various basis sets (Gaussian and plane waves) and with wavefunction methodologies to provide an initial characterization of the groundand excited-state wavefunctions, can be used as a support platform for various types of quantum simulators. The recently developed QDK-NWChem interface 424 (QDK designates Quantum Development Kit developed by Microsoft Research team) for quantum simulations and libraries for CC downfolded electronic Hamiltonians for quantum computing 412 are good illustrations of the utilization of NWChem in supporting the quantum computing effort.…”

“…411 The QDK-NWChem interface with the libDUCC library is used for downfolding electronic Hamiltonians. 412 interact only through these well-defined interfaces and can be combined into full applications. The main motivation is to be able to reuse and swap components as needed and seamlessly create complex applications.…”

Orbital optimized unitary coupled cluster theory for quantum computer

“…VQE calculations using actual quantum computers have already been performed for small molecules [16,[18][19][20][21][22][23]. Recently, researchers have proposed electron-correlation methods based on UCC for quantum computers [24][25][26][27][28][29][30][31][32][33].…”