  In this paper, we consider the ``Shortest Superstring Problem''(SSP) or the
``Shortest Common Superstring Problem''(SCS). The problem is as follows. For a
positive integer $n$, a sequence of n strings $S=(s^1,\dots,s^n)$ is given. We
should construct the shortest string $t$ (we call it superstring) that contains
each string from the given sequence as a substring. The problem is connected
with the sequence assembly method for reconstructing a long DNA sequence from
small fragments. We present a quantum algorithm with running time
$O^*(1.728^n)$. Here $O^*$ notation does not consider polynomials of $n$ and
the length of $t$.
