A hybrid coupled cluster–machine learning algorithm: Development of various regression models and benchmark applications

Abstract

The iterative solution of the coupled cluster equations exhibits a synergistic relationship among the various cluster amplitudes. The iteration scheme is analyzed as a multivariate discrete time propagation of nonlinearly coupled equations, which is dictated by only a few principal cluster amplitudes. These principal amplitudes usually correspond to only a few valence excitations, whereas all other cluster amplitudes are enslaved and behave as auxiliary variables [Agarawal et al., J. Chem. Phys. 154, 044110 (2021)]. We develop a coupled cluster–machine learning hybrid scheme where various supervised machine learning strategies are introduced to establish the interdependence between the principal and auxiliary amplitudes on-the-fly. While the coupled cluster equations are solved only to determine the principal amplitudes, the auxiliary amplitudes, on the other hand, are determined via regression as unique functionals of the principal amplitudes. This leads to significant reduction in the number of independent degrees of freedom during the iterative optimization, which saves significant computation time. A few different regression techniques have been developed, which have their own advantages and disadvantages. The scheme has been applied to several molecules in their equilibrium and stretched geometries, and our scheme, with all the regression models, shows a significant reduction in computation time over the canonical coupled cluster calculations without unduly sacrificing the accuracy.

A Synergistic Approach towards Optimization of Coupled Cluster Amplitudes by Exploiting Dynamical Hierarchy

“…In conjunction to the dynamics of the principal and auxiliary modes shown in Eqs. ( 4) and ( 5), the same for the principal and auxiliary amplitudes in the original (untransformed) orbital basis can be written as: (12) and (13) where, L I and S i are composite hole-particle indices for LAS and SAS elements, respectively. D m is the corresponding orbital energy difference for the general hole-particle index μ.…”

“…Contrarily, there exist large number of other recessive variables (amplitudes) that essentially evolve synergistically as dictated by the dominant ones. The present authors, for the first time, exploited the synchronization among the cluster amplitudes during the optimization trajectory via machine learning that resulted tremendous reduction in the degrees of freedom and 40 %–50 % savings in computational time to achieve sub‐microHartree ($${\mu E_h }$$ ) accuracy [12,13] . As a matter of fact, such a synchronous evolution of all the variables during a (discrete) time evolution is a central theme of Synergetics which gives us a prescription to write the entire dynamics through the collective dominant modes only.…”

“…The present authors, for the first time, exploited the synchronization among the cluster amplitudes during the optimization trajectory via machine learning that resulted tremendous reduction in the degrees of freedom and 40 %-50 % savings in computational time to achieve sub-microHartree (mE h ) accuracy. [12,13] As a matter of fact, such a synchronous evolution of all the variables during a (discrete) time evolution is a central theme of Synergetics which gives us a prescription to write the entire dynamics through the collective dominant modes only. Here we briefly explain the concept of nonlinear dynamics and synergetics and how one may apply the concepts of dimensionality reduction for the CC optimization strategy.…”

Exploring Parameter Redundancy in the Unitary Coupled-Cluster Ansätze for Hybrid Variational Quantum Computing

“…Furthermore, one might anticipate the existence of other hidden relations among the remaining parameters. A way to explore such hidden relations has been reported recently (see refs and ), where it has been observed through phase analysis that the coupled-cluster amplitudes could be divided into principal and auxiliary categories and that a synergistic relationship between them exists during the iteration dynamics. Regarding the UCCSD ansatz, we conjecture that there is a similar significance of the principal amplitudes in the UCC-VQE iterations.…”

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

“…The updated auxiliary modes, however, are fed back to the equations of the principal modes, giving rise to a circularly causal relationship. By determining a functional dependence through which the principal amplitudes are mapped onto the auxiliary amplitudes we can condense the problem only in terms of the former [33][34][35][36].…”