The random k-SAT instances undergo a "phase transition" from being generally satisfiable to unsatisfiable as the clause number m passes a critical threshold, $r_k n$. This causes a drastic reduction in the number of satisfying assignments, shifting the problem from being generally solvable on classical computers to typically insolvable. Beyond this threshold, it is challenging to comprehend the computational complexity of random k-SAT. In quantum computing, Grover's search still yields exponential time requirements due to the neglect of structural information. Leveraging the structure inherent in search problems, we propose the k-local quantum search algorithm, which extends quantum search to structured scenarios. Grover's search, by contrast, addresses the unstructured case where k=n. Given that the search algorithm necessitates the presence of a target, we specifically focus on the problem of searching the interpretation of satisfiable instances of k-SAT, denoted as max-k-SSAT. If this problem is solvable in polynomial time, then k-SAT can also be solved within the same complexity. We demonstrate that, for small $k \ge 3$, any small $\epsilon>0$ and sufficiently large n: $\cdot$ k-local quantum search achieves general efficiency on random instances of max-k-SSAT with $m=\Omega(n^{2+\delta+\epsilon})$ using $\mathcal{O}(n)$ iterations, and $\cdot$ k-local adiabatic quantum search enhances the bound to $m=\Omega(n^{1+\delta+\epsilon})$ within an evolution time of $\mathcal{O}(n^2)$. In both cases, the circuit complexity of each iteration is $\mathcal{O}(n^k)$, and the efficiency is assured with overwhelming probability $1 - \mathcal{O}(\mathrm{erfc}(n^{\delta/2}))$. By modifying this algorithm capable of solving all instances of max-k-SSAT, we further prove that max-k-SSAT is polynomial on average when $m=\Omega(n^{2+\epsilon})$ based on the average-case complexity theory.

翻译：随机$k$-SAT实例在子句数$m$超过临界阈值$r_k n$时，会经历从普遍可满足到不可满足的“相变”。这导致可满足解的数量急剧减少，使得问题从经典计算机上普遍可解转变为通常不可解。超越此阈值后，理解随机$k$-SAT的计算复杂性变得极具挑战。在量子计算中，由于忽略结构信息，Grover搜索仍需要指数时间。利用搜索问题固有的结构，我们提出了$k$-局域量子搜索算法，将量子搜索扩展至结构化场景。相比之下，Grover搜索处理的是$k=n$的非结构化情形。鉴于搜索算法需要目标存在，我们特别关注搜索$k$-SAT可满足实例解释的问题，记为max-$k$-SSAT。若该问题可在多项式时间内求解，则$k$-SAT同样可在相同复杂度内解决。我们证明，对于较小的$k \ge 3$、任意小$\epsilon>0$及充分大的$n$：$\cdot$ $k$-局域量子搜索在$m=\Omega(n^{2+\delta+\epsilon})$的随机max-$k$-SSAT实例上，通过$\mathcal{O}(n)$次迭代实现普遍高效；$\cdot$ $k$-局域绝热量子搜索将界限提升至$m=\Omega(n^{1+\delta+\epsilon})$，演化时间为$\mathcal{O}(n^2)$。两种情形中，每次迭代的电路复杂度均为$\mathcal{O}(n^k)$，且高效性以压倒性概率$1 - \mathcal{O}(\mathrm{erfc}(n^{\delta/2}))$得到保证。通过修改该能求解所有max-$k$-SSAT实例的算法，我们进一步基于平均情形复杂性理论证明：当$m=\Omega(n^{2+\epsilon})$时，max-$k$-SSAT在平均意义下是多项式可解的。