We investigate multivariate integration for a space of infinitely differentiable functions $ \mathcal{F}_{s, \boldsymbol{u}} := \{f \in C^\infty [0,1]^s \mid \| f \|_{\mathcal{F}_{s, \boldsymbol{u}} } := \sup_{(\alpha_1, \dots, \alpha_s) \in \mathbb{N}_0^s} \|f^{(\alpha_1, \dots, \alpha_s)}\|_{L^1}/\prod_{j=1}^s u_j^{\alpha_j} 1$ such that
e(n,s)∈O(q(logn)p) holds as
n goes to infinity, where
e(n,s) may depend on
s in an arbitrary way. We prove that accelerating convergence holds for
p=2 for any
u and
s considered. We also study a sufficient condition for accelerating convergence with strong tractability (AC-ST), which means that there exist constants
C>0,
01 independent of
s such that
e(n,s)≤Cq(logn)p holds for all
s and
n. We prove that AC-ST holds if the weights
u decay sufficiently fast. These results come from the convergence and tractability results for the Walsh space into which
Fs,u is embedded.
The paper studies quantum complexity, tractability, and strong tractability for high dimensional multivariate approximation. We study a space of functions important in many applications. A function space is weighted if certain variables are more important than others; the weights show the relative importance of the variables. In an unweighted space all variables are equally important and multivariate approximation is intractable. We want to study when the complexity of multivariate approximation is independent of the number of variables and depends polynomially on 1/E. The main conclusions are: Multivariate approximation on a quantum computer can be solved roughly (1/E)^(1+r) times faster than on a classical computer using randomization. Here, r is a positive parameter that depends on the weights and may be large. This means that the speed-up of quantum over classical computers may be much larger than quadratic. Multivariate approximation on a quantum computer is exponentially faster than on a classical computer with a worst case assurance even if the sum of weights is infinite but a certain power of them is finite. We have designed a quantum algorithm with error at most E that uses about d+log(1/E) qubits. Hence, we have only linear dependence on the dimension d and logarithmic dependence on 1/E. Therefore, for some applications the number of qubits is quite modest.