Monte Carlo Search for Very Hard KSAT Realizations for Use in Quantum Annealing

Using powerful Multicanonical Ensemble Monte Carlo methods from statistical physics we explore the realization space of random K satisfiability (KSAT) in search for computational hard problems, most likely the 'hardest problems'. We search for realizations with unique satisfying assignments (USA) at ratio of clause to spin number $\alpha=M/N$ that is minimal. USA realizations are found for $\alpha$-values that approach $\alpha=1$ from above with increasing number of spins $N$. We consider small spin numbers in $2 \le N \le 18$. The ensemble mean exhibits very special properties. We find that the density of states of the first excited state with energy one $\Omega_1=g(E=1)$ is consistent with an exponential divergence in $N$: $\Omega_1 \propto {\rm exp} [+rN]$. The rate constants for $K=2,3,4,5$ and $K=6$ of KSAT with USA realizations at $\alpha=1$ are determined numerically to be in the interval $r=0.348$ at $K=2$ and $r=0.680$ at $K=6$. These approach the unstructured search value ${\rm ln}2$ with increasing $K$. Our ensemble of hard problems is expected to provide a test bed for studies of quantum searches with Hamiltonians that have the form of general Ising models.