What does this equation mean?

In 2009, Aram Harrow, Avinatan Hassidim, and Seth Lloyd published “Quantum Algorithm for Linear Systems of Equations” in Physical Review Letters . The problem they addressed — given a system A$\vec{x}$ = $\vec{b}$ , estimate some property of the solution $\vec{x}$ rather than compute $\vec{x}$ itself in full — is not framed as a machine-learning problem in the paper. But it is one of the most common subroutines inside machine-learning problems: linear regression, principal component analysis, and support vector machines all reduce, at some stage, to solving or approximating a linear system. Harrow, Hassidim, and Lloyd showed that a quantum computer could estimate certain expectation values built from $\vec{x}$ in time that scales as $\text{poly}(\log N, \kappa)$ , an exponential improvement in the matrix dimension N over the best classical algorithms available at the time, which scale roughly as N$\sqrt{\kappa}$ for a matrix with condition number $\kappa$ [ 6 ] . What became known as the HHL algorithm is now routinely cited as the theoretical seed that later, explicitly labeled “quantum machine learning” proposals built their linear-algebra subroutines on top of.