Local Quantum Search Algorithm for Random $k$-SAT with $\Omega(n^{1+\epsilon})$ Clauses

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.