Adapting the HHL algorithm to quantum many-body theory

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.