Computation in Finitary Stochastic and Quantum Processes

We introduce stochastic and quantum finite-state transducers as representations of classical stochastic and quantum finitary processes. Formal process languages serve as the literal representation of the behavior of these processes and are recognized and generated by subclasses of stochastic and quantum transducers. We compare deterministic and nondeterministic stochastic and quantum automata, summarizing their relative computational power in a hierarchy of finitary process languages. Quantum finite-state transducers and generators represent a first step toward a computational description of individual closed quantum systems observed over time. They are explored via several physical examples, including the iterated beam splitter, an atom in a magnetic field, and atoms in an ion trap—a special case of which implements the Deutsch quantum algorithm. We show that the behavior of these systems, and so their information processing capacity, depends sensitively on the measurement protocol.