The approximation of weakly singular integral operators plays a key role again. In some situations, techniques from [@Hei05a; @Hei05b] can also be applied to the quantum setting, while in others entirely different approaches are needed. In particular, a number of new tools for the general quantum setting of information-based complexity has to be developed.

The paper is organized as follows. In section 2 we describe the quantum setting, general results about quantum $n$-th minimal errors are derived in section 3. In section 4 we study weighted mean computation and integration. These are preparations for section 5, in which we are concerned with quantum approximation of weakly singular operators. Section 6 contains the statement and the proof of the main result about the query complexity of elliptic PDE. Finally, in section 7 we recall the respective results of the classical deterministic and randomized settings and compare them with the quantum setting.

Notation
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A numerical problem is given by a tuple $\mathcal{P}=(F,G,S,K,\Lambda)$, where $F$ is a non-empty set, $G$ a normed space over ${{\mathbb K}}$, where ${{\mathbb K}}$ stands for the set of real or complex numbers, $S$ a mapping from $F$ to $G$, $K$ a non-empty set and $\Lambda$ a 