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arXiv:0902.2081 · Quantum Information and Computation

Languages recognized by nondeterministic quantum finite automata

Abuzer Yakaryılmaz, A. C. Cem Say

Published Sep 1, 2010• 49 citations
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Abstract

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.

DOI
10.26421/qic10.9-10-3

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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.
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 this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any /spl epsiv/>0, there is a 2qfa M which recognizes the non-regular language L={a/sup m/b/sup m/|m/spl ges/1} with (one-sided) error bounded by E, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1-/spl epsiv/. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfar's which recognize the non-context-free language {a/sup m/b/sup m/c/sup m/|m/spl ges/1}, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular languages. Since it is known that 2-way deterministic, nondeterministic and polynomial expected time, bounded error probabilistic finite automata can recognize only regular languages, it follows that 2qfa's are strictly more powerful than these "classical" models. In the case of 1-way automata, the situation is reversed. We prove that the class of languages recognizable by bounded error 1qfa's is properly contained in the class of regular languages.
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 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 computational model of Quantum Finite Automata has been introduced by multiple authors (e.g. [38, 44]) with some variations in definition. The objective of this thesis is to understand what class of languages can be recognized by these different variations, and how many states are required. We begin by showing that we can use algebraic automata theory to characterize the language recognition power of QFAs. Algebraic automata theory associates to each language a canonical syntactic monoid, and the algebraic structure of this monoid becomes a meaningful parameter in describing language classes. We show that the class of languages recognized by Latvian QFAs [3] corresponds exactly to boolean combinations of languages recognized by Brodsky and Pippenger's QFA model [20], which correspond exactly to those languages whose syntactic monoid is in the class BG. Known results give us a decision procedure for testing membership in this language class. We also use algebraic automata theory to give nearly tight upper and lower bounds on the class of languages recognized by Brodsky and Pippenger's QFAs. We then extend a number of lower bound techniques known for Kondacs and Watrous' 1-way QFA model to Nayak's Generalized QFA. Both of these models are related in that they are permitted to halt before reading the entire input, allowing them to recognize certain languages whose syntactic monoid lies outside of BG. Finally, we investigate the question of QFA succinctness. It is known that QFAs can recognize some languages using exponentially fewer states compared to deterministic finite automata. We extend results from [16] to show that the word problem over abelian groups has this property. We also give example of interesting noncommutative languages with this property.