Carbon monoxide molecules were arranged in atomically precise configurations, which we call "molecule cascades," where the motion of one molecule causes the subsequent motion of another, and so on in a cascade of motion similar to a row of toppling dominoes. Isotopically pure cascades were assembled on a copper (111) surface with a low-temperature scanning tunneling microscope. The hopping rate of carbon monoxide molecules in cascades was found to be independent of temperature below 6 kelvin and to exhibit a pronounced isotope effect, hallmarks of a quantum tunneling process. At higher temperatures, we observed a thermally activated hopping rate with an anomalously low Arrhenius prefactor that we interpret as tunneling from excited vibrational states. We present a cascade-based computation scheme that has all of the devices and interconnects required for the one-time computation of an arbitrary logic function. Logic gates and other devices were implemented by engineered arrangements of molecules at the intersections of cascades. We demonstrate a three-input sorter that uses several AND gates and OR gates, as well as the crossover and fan-out units needed to connect them.