$C(Q)$. We consider the problem $\mathcal{P}=(B_{C(Q)},{{\mathbb K}}, I_{Q,g},{{\mathbb K}}, \Lambda)$ with $\Lambda=\{\delta_x:\, x\in Q\}$, where $\delta_x(f)=f(x)$ for $f\in C(Q)$.

\[pro:Q2\] There is a constant $c>0$ such that for each closed, bounded set $Q\subset {{\mathbb R}}^d$ of positive Lebesgue measure, for all $g\in L_1(Q)$ and $n\in {{\mathbb N}}$ $$\sup_{\mathcal{E}\in\mathscr{C}( B_{C(Q)})}
e_n^{{\rm q }}(I_{Q,g},\mathcal{E})\le cn^{-1}\|g\|_{L_1(Q)}.$$

[**Remark.**]{} It is well-known and easily checked by using importance sampling with density function $|g|/\|g\|_{L_1(Q)}$ that in the classical randomized setting we have $$e_n^{{\rm ran}}(I_{Q,g},B_{C(Q)})\le cn^{-1/2}\|g\|_{L_1(Q)},$$ where $e_n^{{\rm ran}}$ is the $n$-th minimal classical randomized error (see, e.g., [@Hei05b], section 3). Proposition \[pro:Q2\] is the quantum analogue of this result. Let us comment on the reasons for taking the supremum over $\mathcal{E}\in\mathscr{C}( B_{C(Q)})$. In contrast to the classical randomized setting, no non-trivial convergence rate holds for $e_n^{{\rm q }}(I_{Q,g},B_{C(Q)})$, in general. This is easily checked based on the fact that a quantum query involves, by definition, the values of functions from $B_{C(Q)}$ in a finite set of points of $Q$ only. For situations like this a natural way of formulating quantum counterparts of results of the classical randomized setting was already observed in section 5 of [@Hei01]: If we restrict our analysis to uniformly equicontinuous subsets $\mathcal{E}$ 