Fast Quantum Algorithm for Numerical Gradient Estimation

Given a black box for f, a smooth real scalar function of d real variables, one wants to estimate ∇f at a given point with n bits of precision. On a classical computer this requires a minimum of d+1 black box queries, whereas on a quantum computer it requires only one query regardless of d. The number of bits of precision to which f must be evaluated matches the classical requirement in the limit of large n.