Quantum algorithm for solving linear systems of equations

TL;DR

This paper presents a quantum algorithm that efficiently estimates the expectation value of a quadratic form related to the solution of a linear system, offering an exponential speedup over classical methods for sparse matrices.

Contribution

The authors introduce a quantum algorithm capable of estimating expectation values of solutions to linear systems with exponential speedup for sparse matrices.

Findings

Quantum algorithm runs in poly(log N, kappa) time

Classical algorithms require O(N sqrt(kappa)) time

Quantum approach significantly reduces computational complexity

Abstract

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.