In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)
In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)
Abstract. We prove that the class of languages recognized by generalized quantum finite automata (GQFA’s) with unbounded error equals the class of stochastic languages. The capability of performing additional intermediate projective measurements does not increase the recognition power of GQFA’s over Kondacs-Watrous quantum finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input enlarges the class of languages recognized by GQFA’s with unbounded error. 1
Abstract. We prove that several variants of the quantum finite automaton (QFA) model are strictly superior to their probabilistic counterparts, in terms of both language recognition power and succinctness, in the onesided unbounded error setting. This is the first case where one-way QFA models have been shown to outperform the corresponding probabilistic automata in both these respects. We give a characterization of the class of languages recognized by these QFA’s with cutpoint 0, demonstrating that it is equal to the class of exclusive stochastic languages. We prove several closure properties of the related classes. 1
In automata theory, quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to QFAs augmented with counters or stacks. In this paper, we focus on such generalizations of QFAs where the input head operates in one-way or realtime mode, and present some new results regarding their superiority over their classical counterparts. Our first result is about the nondeterministic acceptance mode: Each quantum model architecturally intermediate between realtime finite state automaton and one-way pushdown automaton (one-way finite automaton, realtime and one-way finite automata with one-counter, and realtime pushdown automaton) is superior to its classical counterpart. The second and third results are about bounded error language recognition: for any k > 0, QFAs with k blind counters outperform their deterministic counterparts; and, a one-way QFA with a single head recognizes an infinite family of languages, which can be recognized by one-way probabilistic finite automata with at least two heads. Lastly, we compare the nondeterminictic and deterministic acceptance modes for classical finite automata with k blind counter(s), and we show that for any k > 0, the nondeterministic models outperform the deterministic ones.
The nondeterministic quantum finite automaton (NQFA) is the only known case where a one-way quantum finite automaton (QFA) model has been shown to be strictly superior in terms of language recognition power to its probabilistic counterpart. We give a characterization of the class of languages recognized by NQFAs, demonstrating that it is equal to the class of exclusive stochastic languages. We also characterize the class of languages that are recognized necessarily by two-sided error by QFAs. It is shown that these classes remain the same when the QFAs used in their definitions are replaced by several different model variants that have appeared in the literature. We prove several closure properties of the related classes. The ramifications of these results about classical and quantum sublogarithmic space complexity classes are examined.
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.
The 2-way quantum finite automaton introduced by Kondacs and Watrous [Proceedings of the 38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75] can accept nonregular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1-way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1-way quantum finite automata. The first model, termed measure-once quantum finite automata, was introduced by Moore and Crutchfield [Theoret. Comput. Sci., 237 (2000), pp. 275--306], and the second model, termed measure-many quantum finite automata, was introduced by Kondacs and Watrous [Proceedings of the38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75]. We characterize the measure-once model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measure-once automata are equivalent. We prove several closure properties of the classes of languages accepted by measure-many automata, including inverse homomorphisms, and provide a new necessary condition for a language to be accepted by the measure-many model with bounded error. Finally, we show that piecewise testable sets can be accepted with bounded error by a measure-many quantum finite automaton, introducing new construction techniques for quantum automata in the process.