We develop an analytical theory for quantum phase transitions driven by disorder in magnets and superconductors. We study these transitions with a cavity approximation which becomes exact on a Bethe lattice with large branching number. We find two di
fferent disordered phases, characterized by very different relaxation rates, which both exhibit strong inhomogeneities typical of glassy physics.
We study the phase diagram and the algorithmic hardness of the random `locked constraint satisfaction problems, and compare them to the commonly studied non-locked problems like satisfiability of boolean formulas or graph coloring. The special proper
ty of the locked problems is that clusters of solutions are isolated points. This simplifies significantly the determination of the phase diagram, which makes the locked problems particularly appealing from the mathematical point of view. On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. Our results suggest that the easy/hard transition (for currently known algorithms) in the locked problems coincides with the clustering transition. These should thus be regarded as new benchmarks of really hard constraint satisfaction problems.
We introduce and study the random locked constraint satisfaction problems. When increasing the density of constraints, they display a broad clustered phase in which the space of solutions is divided into many isolated points. While the phase diagram
can be found easily, these problems, in their clustered phase, are extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. We thus propose new benchmarks of really hard optimization problems and provide insight into the origin of their typical hardness.
