On the quantum and randomized approximation of linear functionals on function spaces

Abstract

We deal with quantum and randomized algorithms for approximating a class of linear continuous functionals. The functionals are defined on a Hölder space of functions f of d variables with r continuous partial derivatives, the rth derivative being a Hölder function with exponent ρ. For a certain class of such linear problems (which includes the integration problem), we define algorithms based on partitioning the domain of f into a large number of small subdomains, and making use of the well-known quantum or randomized algorithms for summation of real numbers. For N information evaluations (quantum queries in the quantum setting), we show upper bounds on the error of order N ^−(γ+1) in the quantum setting, and N ^−(γ+1/2) in the randomized setting, where γ = (r + ρ)/d is the regularity parameter. Hence, we obtain for a wider class of linear problems the same upper bounds as those known for the integration problem. We give examples of functionals satisfying the assumptions, among which we discuss functionals defined on the solution of Fredholm integral equations of the second kind, with complete information about the kernel. We also provide lower bounds, showing in some cases sharpness of the obtained results, and compare the power of quantum, randomized and deterministic algorithms for the exemplary problems.