  Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum
computers may have an exponential advantage in solving a wealth of linear
algebraic problems, over classical algorithms. Building on the quantum
intuition of these results, we step back into the classical domain, and explore
its usefulness in designing classical algorithms. We achieve an algorithm for
solving the major linear-algebraic problems in time $O(n^{\omega+\nu})$ for any
$\nu>0$, where $\omega$ is the optimal matrix-product constant. Thus our
algorithm is optimal w.r.t. matrix multiplication, and comparable to the
state-of-the-art algorithm for these problems due to Demmel et. al. Being
derived from quantum intuition, our proposed algorithm is completely disjoint
from all previous classical algorithms, and builds on a combination of
low-discrepancy sequences and perturbation analysis. As such, we hope it
motivates further exploration of quantum techniques in this respect, hopefully
leading to improvements in our understanding of space complexity and numerical
stability of these problems.
