  Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential
speedup in system size for treating linear equations of the form
$A\vec{x}=\vec{b}$ on quantum computers when compared to their traditional
counterparts, it faces a challenge related to the condition number
($\mathcal{\kappa}$) scaling of the $A$ matrix. In this work, we address the
issue by introducing the post-selection-improved HHL (Psi-HHL) approach that
involves a simple yet effective modification of the HHL algorithm to extract a
feature of $|x\rangle$, and which leads to achieving optimal behaviour in
$\mathcal{\kappa}$ (linear scaling) for large condition number situations. This
has the important practical implication of having to use substantially fewer
shots relative to the traditional HHL algorithm. We carry out two sets of
simulations, where we go up to 26-qubit calculations, to demonstrate the
ability of Psi-HHL to handle situations involving large $\mathcal{\kappa}$
matrices via: (a) a set of toy matrices, for which we go up to size $64 \times
64$ and $\mathcal{\kappa}$ values of up to $\approx$ 1 million, and (b) a
deep-dive into quantum chemistry, where we consider matrices up to size $256
\times 256$ that reach $\mathcal{\kappa}$ of about 466. The molecular systems
that we consider are Li$_{\mathrm{2}}$, RbH, and CsH. Although the feature of
$|x\rangle$ considered in our examples is an overlap between the input and
output states of the HHL algorithm, our approach is general and can be applied
in principle to any transition matrix element involving $|x\rangle$.
