Randomized and quantum complexity of nonlinear two-point BVPs

摘要

We deal with the complexity of nonlinear BVPs with nonlinear two-point boundary conditions. We consider the randomized and quantum models of computation. We assume that the right-hand side function is r times differentiable with all derivatives bounded by a constant. We show that the ε-complexity is roughly of order ε-1/(r+1/2) in the randomized setting, and ε-1/(r+1) in the quantum setting. We compare our results with known results in the deterministic setting. The speed-up of the randomized computations with respect to the deterministic computations is by 1/(r(2r+1)) in the exponent of 1/ε, and the speed-up of the quantum computations by 1/(r(r+1)) in the exponent.