Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers

Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form Avec{x}=vec{b} on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number $mathcalκ$ scaling of the A matrix. In this work, we address the issue by introducing the post-selection-improved HHL (Psi-HHL) framework that operates on a simple yet effective premise: subtracting mixed and wrong signals to extract correct signals while providing the benefit of optimal scaling in the condition number of A denoted as $mathcalκ$ for large mathcalκ scenarios. This approach, which leads to minimal increase in circuit depth, has the important practical implication of having to use substantially fewer shots relative to the traditional HHL algorithm. The term `signal' refers to a feature of |xrangle. We design circuits for overlap and expectation value estimation in the Psi-HHL framework. We demonstrate performance of Psi-HHL via numerical simulations. We carry out two sets of computations, where we go up to 26-qubit calculations, to demonstrate the ability of Psi-HHL to handle situations involving large mathcalκ matrices via: (a) a set of toy matrices, for which we go up to size 64 times 64 and mathcalκ values of up to approx 1 million, and (b) application to quantum chemistry, where we consider matrices up to size 256 times 256 that reach mathcalκ of about 393. The molecular systems that we consider are Li₂, KH, RbH, and CsH.