Tight bound for estimating expectation values from a system of linear equations

The system of linear equations problem (SLEP) is specified by a complex invertible matrix A, the condition number kappa of A, a vector b, a Hermitian matrix M, and an accuracy epsilon, and the task is to estimate x(dagger)Mx, where x is the solution vector to the equation Ax = b. We aim to establish a lower bound on the complexity of the end-to-end quantum algorithms for SLEP with respect to epsilon, and devise a quantum algorithm that saturates this bound. To make lower bounds attainable, we consider query complexity in the setting in which a block encoding of M is given, i.e., a unitary black box U-M that contains M/alpha as a block for some alpha is an element of R+. We show, by constructing a quantum algorithm and deriving a lower bound, that the quantum query complexity for SLEP in this setting is Theta(alpha/epsilon). Our lower bound is established by reducing the problem of estimating the mean of a black box function to SLEP. Our Theta(alpha/epsilon) result tightens and proves the common assertion of polynomial accuracy dependence [poly(1/epsilon)] for SLEP without making any complexity-theoretic assumptions, and shows that improvement beyond linear dependence on accuracy is not possible if M is provided via block encoding.