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.

翻译：随机$k$-SAT问题的计算复杂度取决于子句数量$m$。在经典计算中，可满足性阈值在$m=r_k n$处被确立，标志着随机$k$-SAT从可满足到不可满足的转变。然而，超出这一既定阈值，理解其复杂度仍具挑战性。在量子计算机上，直接应用Grover非结构化量子搜索因忽略结构信息仍需指数级时间。本文提出一类名为$k$-局部量子搜索的结构化量子搜索算法，专门用于解决$k$-SAT问题。由于搜索算法需存在目标，我们聚焦于$k$-SAT的可满足侧，即可满足实例上的max-$k$-SAT（记为max-$k$-SSAT），其中$k \ge 3$较小。对于$m=\Omega(n^{2+\epsilon})$的随机实例，当任意小$\epsilon>0$且$n$足够大时，证明了通用的指数加速。此外，绝热$k$-局部量子搜索将通用效率的界限改进至$m=\Omega(n^{1+\epsilon})$，进化时间为$\mathcal{O}(n^2)$。特别地，当$m=\Theta(n^{1+\delta+\epsilon})$时，算法在概率$1-\mathcal{O}(\mathrm{erfc}(n^{\delta/2}))$下保证效率。通过修改此算法使其能够求解所有实例，基于平均复杂度理论，我们证明当$m=\Omega(n^{2+\epsilon})$时，max-$k$-SSAT在平均意义上是多项式可解的。