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 and a vector , find a vector such that . We consider the case where one does not need to know the solution itself, but rather an approximation of the expectation value of some operator associated with , e.g., for some matrix . In this case, when is sparse, and has condition number , the fastest known classical algorithms can find and estimate in time scaling roughly as . Here, we exhibit a quantum algorithm for estimating whose runtime is a polynomial of and . Indeed, for small values of [i.e., ], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.