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

Abstract

The coupled cluster iteration scheme for determining the cluster amplitudes involves a set of nonlinearly coupled difference equations. In the space spanned by the amplitudes, the set of equations are analyzed as a multivariate time-discrete map where the concept of time appears in an implicit manner. With the observation that the cluster amplitudes have difference in their relaxation timescales with respect to the distributions of their magnitudes, the coupled cluster iteration dynamics are considered as a synergistic motion of coexisting slow and fast relaxing modes, manifesting a dynamical hierarchical structure. With the identification of the highly damped auxiliary amplitudes, their time variation can be neglected compared to the principal amplitudes which take much longer time to reach the fixed points. We analytically establish the adiabatic approximation where each of these auxiliary amplitudes are expressed as unique parametric functions of the collective principal amplitudes, allowing us to study the optimization with the latter taken as the independent degrees of freedom. Such decoupling of the amplitudes significantly reduces the computational scaling without sacrificing the accuracy in the ground state energy as demonstrated by a number of challenging molecular applications. A road-map to treat higher order post-adiabatic effects is also discussed.

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

“…In other words, it is only these small number of parameters chosen in the described manner that contribute in a major capacity towards the phase space trajectory traversed during the PQE iterative scheme. This particular behaviour corroborates the existence of two distinct classes of parameters present [33,34,36] 1. Principal Parameter Subset (PPS ): A relatively smaller subset {θ P } consisting of parameters that have larger absolute magnitudes and longer relaxation timescale to reach the fixed point.…”

“…The summations in equation Eq. ( 19) can be readily performed, resulting in the final expression [36,[51][52][53]…”

“…( 12) as the adiabatic approximation [32] suggests. To obtain a fully decoupled set of equations only for the auxiliary variables we can approximately neglect [34,36,54] all the θ A that are present in M ({θ P , θ A }) with respect to θ P due to their relative magnitude -…”

“…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].…”

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

“…Regarding the UCCSD ansatz, we conjecture that there is a similar significance of the principal amplitudes in the UCC-VQE iterations. Although obtaining a general and rigorous mathematical proof is more convoluted, , a numerical phase space trajectory associated with the iterations can be straightforwardly observed for the test cases. For example, as shown in Figure , for LiH, the distance matrices computed with full UCCSD amplitudes and with only principal UCCSD amplitudes are nearly identical, indicating that the principal amplitudes can replicate the evolution exhibited by the full set of amplitudes.…”