Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice
Tomoyuki Yamakami
TL;DR
The paper develops a comprehensive framework linking nonuniform state complexity of finite automata, including quantum variants with garbage tapes, to space-bounded and parameterized complexity with advice. It defines a spectrum of nonuniform quantum/probabilistic classes (e.g., $1\mathrm{BQ}$, $2\mathrm{BQ}$, $2\mathrm{Q}$) and proves a network of inclusions and separations, with crucial results connecting advised, log-space quantum computation to nonuniform automata. A central contribution is a theorem relating $\mathrm{ptime-}2\mathcal{A}/\mathrm{poly}$ to parameterized classes like $\mathrm{para-}\mathrm{L}/\mathrm{poly}$ and $\mathrm{para-}\mathrm{NL}/\mathrm{poly}$, via translations between QTMs and 2qfas, aided by quantum transition tables and quantum advice. This unifies nonuniform automata theory with parameterized/advised quantum computation, yielding structural insights and paving the way for further exploration of quantum/nonuniform models and their practical implications.
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 the nonuniform complexity classes defined by such families of finite automata having polynomial-size state complexity. In a similar fashion, we introduce nonuniform state complexity classes using nonuniform families of quantum finite automata empowered by the flexible use of garbage tapes. We first present general inclusion and separation relationships among nonuniform state complexity classes of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata having polynomially many inner states. For two-way quantum finite automata equipped with flexible garbage tapes, we show a close relationship between the nonuniform state complexity of the family of such polynomial-size quantum finite automata and the parameterized complexity class induced by logarithmic-space quantum computation assisted by polynomial-size advice. We further establish a direct connection between space-bounded quantum computation with quantum advice and quantum finite automata whose transitions are dictated by superpositions of transition tables.