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

TL;DR

This paper introduces two quantum algorithms, including a variable time amplitude amplification method and an improved algorithm for solving linear systems, significantly reducing computational complexity.

Contribution

It presents a novel variable time amplitude amplification technique and enhances the quantum linear system solver with faster running time.

Findings

Improved quantum algorithm for linear systems with reduced complexity

Generalization of amplitude amplification for variable stopping times

Faster solution of linear equations with complexity O(kappa log^3 kappa log N)

Abstract

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(kappa^2 log N) to O(kappa log^3 kappa log N) where \kappa is the condition number of the system of equations.