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 which are believed to be intractable for classical and quantum computers, respectively. 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 new 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.Comment: Updated reference

A Computational Model for Quantum Measurement

Is the dynamical evolution of physical systems objectively a manifestation of information processing by the universe? We find that an affirmative answer has important consequences for the measurement problem. In particular, we calculate the amount of quantum information processing involved in the evolution of physical systems, assuming a finite degree of fine-graining of Hilbert space. This assumption is shown to imply that there is a finite capacity to sustain the immense entanglement that measurement entails. When this capacity is overwhelmed, the system's unitary evolution becomes computationally unstable and the system suffers an information transition (`collapse'). Classical behaviour arises from the rapid cycles of unitary evolution and information transitions. Thus, the fine-graining of Hilbert space determines the location of the `Heisenberg cut', the mesoscopic threshold separating the microscopic, quantum system from the macroscopic, classical environment. The model can be viewed as a probablistic complement to decoherence, that completes the measurement process by turning decohered improper mixtures of states into proper mixtures. It is shown to provide a natural resolution to the measurement problem and the basis problem.Comment: 24 pages; REVTeX4; published versio