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

Abstract

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 $α=M/N$ that is minimal. USA realizations are found for $α$-values that approach $α=1$ 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 $Ω_1=g(E=1)$ is consistent with an exponential divergence in $N$: $Ω_1 \propto {\rm exp} [+rN]$. The rate constants for $K=2,3,4,5$ and $K=6$ of KSAT with USA realizations at $α=1$ 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.