Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations

We present two new quantum algorithms. Our first algorithm is a generalization of amplitude amplification to the case when parts of the quantum algorithm that is being amplified stop at different times. Our second algorithm uses the first algorithm to improve the running time of Harrow et al. algorithm for solving systems of linear equations from O(κ logN) to O(κ log κ logN) where κ is the condition number of the system of equations.