While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an \(n\)-dimensional convex body using \(\tilde{O}(n)\) queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires \(\tilde{\Omega}(\sqrt n)\) evaluation queries and \(\Omega(\sqrt{n})\) membership queries.