any quantum 
algorithm with}\,\, e(S,A,F) \le \varepsilon\}\end{aligned}$$ (we put $\mbox{comp}_\varepsilon^{{\rm q }}(S,F)=+\infty$ if there is no such algorithm). It is easily checked that these functions are inverse to each other in the following sense: For all $n\in {{\mathbb N}}_0$ and $\varepsilon > 0$, $e_n^{{\rm q }}(S,F)\le \varepsilon$ if and only if $\mbox{comp}_{\varepsilon_1}^{{\rm q }}(S,F)\le n$ for all $\varepsilon_1 > \varepsilon$. Hence it suffices to determine one of them. We shall principally choose the first one.

Note that the definition of a numerical problem we presented here corresponds to that used in [@Hei05a; @Hei05b] for the classical settings, and is slightly more general than the one in previous papers on the quantum setting [@Hei01; @Hei01b; @Hei04a; @Hei04b]. There $F$ was always a set of functions on some set $D$. We get back to this setting by considering, as done above, each $f$ as a function on $\Lambda$ and defining $D=\Lambda$. (Such an approach has already been outlined at the end of [@Hei01].) The mapping that sends $f\in F$ to the corresponding function $(f({{\lambda }}))_{{{\lambda }}\in \Lambda}$ needs not to be one-to-one, in general. Nevertheless, all general results of [@Hei01; @Hei01b; @Hei04a; @Hei04b] carry over in an obvious way, with literally identical 