  We provide a method for estimating the expectation value of an operator that
can utilize prior knowledge to accelerate the learning process on a quantum
computer. Specifically, suppose we have an operator that can be expressed as a
concise sum of projectors whose expectation values we know a priori to be
$O(\epsilon)$. In that case, we can estimate the expectation value of the
entire operator within error $\epsilon$ using a number of quantum operations
that scales as $O(1/\sqrt{\epsilon})$. We then show how this can be used to
reduce the cost of learning a potential energy surface in quantum chemistry
applications by exploiting information gained from the energy at nearby points.
Furthermore, we show, using Newton-Cotes methods, how these ideas can be
exploited to learn the energy via integration of derivatives that we can
estimate using a priori knowledge. This allows us to reduce the cost of energy
estimation if the block-encodings of directional derivative operators have a
smaller normalization constant than the Hamiltonian of the system.
