Quantum Complexity of Integration

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F^k, α_d on 0, 1]^d and define γ by γ=(k+α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are comp(F^k, α_d, ε)≍ε^−1/γ and comp^random(F^k, α_d, ε)≍ε^−2/(1+2γ). For a quantum computer we prove comp^quant_query(F^k, α_d, ε)≍ε^−1/(1+γ) and comp^quant(F^k, α_d, ε)⩽Cε^−1/(1+γ)(log ε⁻¹)^1/(1+γ). For restricted Monte Carlo (only coin tossing instead of general random numbers) we prove comp^coin(F^k, α_d, ε)⩽Cε^−2/(1+2γ)(log ε⁻¹)^1/(1+2γ). To summarize the results one can say that   • there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if γ is small;   • there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if γ is small.