Quantum algorithm for linear systems of equations

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 does not 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×N and has condition number κ, the fastest known classical algorithms can find x→ and estimate x→†Mx→ in time scaling roughly as Nκ. Here, we exhibit a quantum algorithm for estimating x→†Mx→ whose runtime is a polynomial of log(N) and κ. Indeed, for small values of κ [i.e., polylog(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.