Classical-quantum mixing in the random 2-satisfiability problem

Classical satisfiability (SAT) and quantum satisfiability (QSAT) are complete problems for the complexity classes NP and QMA, respectively, and they are believed to be intractable for both classical and quantum computers. Statistical ensembles of instances of these problems have been studied previously in an attempt to elucidate their typical, as opposed to worst-case, behavior. In this paper, we introduce a statistical ensemble that interpolates between classical and quantum. For the simplest 2-SAT-2-QSAT ensemble, we find the exact boundary that separates SAT and UNSAT instances. We do so by establishing coincident lower and upper bounds, in the limit of large instances, on the extent of the UNSAT and SAT regions, respectively.