identify ${{\mathbb Z}}[0,2^m)$ with $\{0,1\}\times{{\mathbb Z}}[0,2^{m''})$. Define $\bar{{\varphi }}:({{\mathbb Z}}[0,2^m))^n\to G$ for $$(b_i,a_i)\in \{0,1\}\times{{\mathbb Z}}[0,2^{m''})={{\mathbb Z}}[0,2^m)\quad(i=0,\dots,n-1)$$ by setting $$\bar{{\varphi }}((b_0,a_0), \dots, (b_{n-1},a_{n-1}))={\varphi }({\gamma }(a_0), \dots,{\gamma }(a_{n-1})).$$ Now we construct a quantum algorithm $A$ with $n$ measurements. We let $$A=((A_l)_{l=0}^{n-1},(b_l)_{l=0}^{n-1},\bar{\varphi}),$$ with $b_0=0$, $b_l\equiv 0\;(1\le l\le n-1)$, and $\bar{{\varphi }}$ as above. Each $A_l$ is of the form $$A_l=(Q_l,(U_{lj})_{j=0,1})$$ with $U_0=U_1=I_{H_m}$ the identity matrix, $$Q_l=(m,m',m'', Z_l,\tau,{\beta }),$$ with $m,m',m'',{\beta }$ as defined above, $Z_l=\{0\}$ and $\tau_l(0)={{\lambda }}_l$ $(l=0,\dots,n-1)$. This simply means that $A$ is an algorithm which queries the function $f$ in the appropriate $n$ points, with the needed precision, and measures the result after each query. Finally $\bar{{\varphi }}$ is applied. Let $f\in F$ and let $\zeta$ have distribution $A(f)$. Since the computation remains on the classical states, the measurements give the result $$(0,{\beta }(f({{\lambda }}_0))),\dots,(0,{\beta }(f({{\lambda }}_{n-1})))$$ with probability 1. Hence $$\begin{aligned}
\zeta&=&\bar{{\varphi }}({\beta }\circ f({{\lambda }}_0),\dots,{\beta }\circ f({{\lambda }}_{n-1}))\\
&=&{\varphi }({\gamma }\circ{\beta }\circ f({{\lambda }}_0),\dots,{\gamma }\circ{\beta }\circ f({{\lambda }}_{n-1}))\\
&=&{\varphi }(\theta\circ f({{\lambda }}_0),\dots,\theta\circ f({{\lambda }}_{n-1}))={\widetilde }{A}(\Theta(f))\end{aligned}$$ with probability 1, consequently, by conclusion (\[RU2\]) of Lemma \[lem:2\] and by (\[RU5\]) $$\begin{aligned}
\|S(f)-\zeta\|&\le& \|S(f)-S(\Theta(f))\|+\|S(\Theta(f))-{\widetilde }{A}(\Theta(f))\|\\
&\le& \|S\|\delta +2e_n^{{\rm det}}(S,F)+\delta,\end{aligned}$$ and (\[VU1\]) follows.

From Proposition \[pro:4\] and Lemma \[lem:3\] we immediately conclude

