  We show that exponential sums (ES) of the form \begin{equation*} S(f, N)=
\sum_{k=0}^{N-1} \sqrt{w_k} e^{2 \pi i f(k)}, \end{equation*} can be
efficiently carried out with a quantum computer (QC). Here $N$ can be
exponentially large, $w_k$ are real numbers such that sum
$S_w(M)=\sum_{k=0}^{M-1} w_k$ can be calculated in a closed form for any $M$,
$S_w(N)=1$ and $f(x)$ is a real function, that is assumed to be easily
implementable on a QC. As an application of the technique, we show that Riemann
zeta (RZ) function, $\zeta(\sigma+ i t)$ in the critical strip, $\{0 \le \sigma
<1, t \in \mathbb{R} \}$, can be obtained in polyLog(t) time. In another
setting, we show that RZ function can be obtained with a scaling $t^{1/D}$,
where $D \ge 2$ is any integer. These methods provide a vast improvement over
the best known classical algorithms; best of which is known to scale as
$t^{4/13}$. We present alternative methods to find $\lvert S(f,N) \rvert$ on a
QC directly. This method relies on finding the magnitude $A=\lvert \sum_0^{N-1}
a_k \rvert$ of a $n$-qubit quantum state with $a_k$ as amplitudes in the
computational basis. We present two different ways to do obtain $A$. Finally, a
brief discussion of phase/amplitude estimation methods is presented.
