  Rapid progress in developing near- and long-term quantum algorithms for
quantum chemistry has provided us with an impetus to move beyond traditional
approaches and explore new ways to apply quantum computing to electronic
structure calculations. In this work, we identify the connection between
quantum many-body theory and a quantum linear solver, and implement the
Harrow-Hassidim-Lloyd (HHL) algorithm to make precise predictions of
correlation energies for light molecular systems via the (non-unitary)
linearised coupled cluster theory. We alter the HHL algorithm to integrate two
novel aspects- (a) we prescribe a novel scaling approach that allows one to
scale any arbitrary symmetric positive definite matrix A, to solve for Ax = b
and achieve x with reasonable precision, all the while without having to
compute the eigenvalues of A, and (b) we devise techniques that reduce the
depth of the overall circuit. In this context, we introduce the following
variants of HHL for different eras of quantum computing- AdaptHHLite in its
appropriate forms for noisy intermediate scale quantum (NISQ), late-NISQ, and
the early fault-tolerant eras, as well as AdaptHHL for the fault-tolerant
quantum computing era. We demonstrate the ability of the NISQ variant of
AdaptHHLite to capture correlation energy precisely, while simultaneously being
resource-lean, using simulation as well as the 11-qubit IonQ quantum hardware.
