Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice

Abstract

The state complexity of a finite(-state) automaton intuitively measures the size of the description of the automaton. Sakoda and Sipser [STOC 1972, pp. 275–286] were concerned with nonuniform families of finite automata and they discussed the behaviors of nonuniform complexity classes defined by families of such finite automata having polynomial-size state complexity. In a similar fashion, we introduce nonuniform state complexity classes using families of quantum finite automata. Our primarily concern is one-way quantum finite automata empowered by garbage tapes. We show inclusion and separation relationships among nonuniform state complexity classes of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata of polynomial size. For two-way quantum finite automata equipped with garbage tapes, we discover a close relationship between the nonuniform state complexity of such a polynomial-size quantum finite automata family and the parameterized complexity class induced by quantum logarithmic-space computation assisted by polynomial-size advice.