Quantum linear solvers for quantum chemistry: prospects of exponential quantum advantage

Quantum linear solvers (QLSs) can offer the potential for exponential quantum advantage in solving quantum chemical problems, but its assessment hinges on determining the condition number ($\kappa$) scaling, which itself is computationally challenging.

While a recent work applied the Harrow-Hassidim-Lloyd (HHL) algorithm to single-reference linearized coupled cluster equations (SRLCC), the validity of the HHL-SRLCC framework is restricted to weakly correlated regimes.

A general treatment requires a formulation that can access strongly correlated regions.

We thus begin by extending the QLS-SRLCC framework to its multi-reference form, which is based on the internally contracted multi-reference LCC method (QLS-icMRLCC).

We then analyze $\kappa$ scaling using three complementary diagnostics that range from explicit computations to use of indirect structural indicators: (i) direct calculations of $\kappa$, (ii) scaling of the ratio of maximum to minimum diagonal entries of an A matrix, and (iii) structural analyses of the A matrices based on a recently proposed conjecture, which we adapt to the QLS-LCC problem.

The three approaches yield consistent predictions, indicating a polylogarithmic $\kappa$ scaling in system size.

This finding, when combined with our arguments on sub-linear scaling of sparsity, supports the prospects of exponential advantage using QLSs for the LCC problem.

Finally, numerical calculations on potential energy curves of model systems containing up to four atoms recover the ground state energies with errors relative to benchmark classical methods not exceeding 0.009$\%$.