Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd designed a quantum algorithm for linear systems of equations. The algorithm is fundamental for two reasons: (1) it provides an exponential speedup over the best known classic algorithm, and (2) it also solves a BPQ problem, therefore could be an essential subroutine in many future quantum algorithms. The algorithm is sometimes referred to as the HHL matrix inversion algorithm after it’s designers and off of the fact that it solves $Ax = b$ by essential inverting $A$ and finding $x$ from $x = A-1b$. This presentation will cover a summary of the classic approach, simple view of the quantum approach, runtime, and error analysis of the quantum algorithm.
(Based on a paper by Harrow, Hassidim, and Lloyd in Phys. Rev. Lett. 2009)