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Using powerful Multicanonical Ensemble Monte Carlo methods from statisticalphysics we explore the realization space of random K satisfiability (KSAT) insearch for computational hard problems, most likely the 'hardest problems'. Wesearch for realizations with unique satisfying assignments (USA) at ratio ofclause to spin number $\alpha=M/N$ that is minimal. USA realizations are foundfor $\alpha$-values that approach $\alpha=1$ from above with increasing numberof spins $N$. We consider small spin numbers in $2 \le N \le 18$. The ensemblemean exhibits very special properties. We find that the density of states ofthe first excited state with energy one $\Omega_1=g(E=1)$ is consistent with anexponential divergence in $N$: $\Omega_1 \propto {\rm exp} [+rN]$. The rateconstants 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 atest bed for studies of quantum searches with Hamiltonians that have the formof general Ising models.

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