  The computational complexity of random $k$-SAT problem is contingent on the
clause number $m$. In classical computing, a satisfiability threshold is
identified at $m=r_k n$, marking the transition of random $k$-SAT from
solubility to insolubility. However, beyond this established threshold,
comprehending the complexity remains challenging. On quantum computers, direct
application of Grover's unstructured quantum search still yields exponential
time requirements due to oversight of structural information. This paper
introduces a family of structured quantum search algorithms, termed $k$-local
quantum search, designed to address the $k$-SAT problem. Because search
algorithm necessitates the presence of a target, our focus is specifically on
the satisfiable side of $k$-SAT, i.e., max-$k$-SAT on satisfiable instances,
denoted as max-$k$-SSAT, with a small $k \ge 3$. For random instances with
$m=\Omega(n^{2+\epsilon})$, general exponential acceleration is proven for any
small $\epsilon>0$ and sufficiently large $n$. Furthermore, adiabatic $k$-local
quantum search improves the bound of general efficiency to
$m=\Omega(n^{1+\epsilon})$, within an evolution time of $\mathcal{O}(n^2)$.
Specifically, for $m=\Theta(n^{1+\delta+\epsilon})$, the efficiency is
guaranteed in a probability of $1-\mathcal{O}(\mathrm{erfc}(n^{\delta/2}))$. By
modifying this algorithm capable of solving all instances, we prove that the
max-$k$-SSAT is polynomial on average if $m=\Omega(n^{2+\epsilon})$ based on
the average-case complexity theory.
