We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $ s $ satisfying $ s(n)=o(\log \log n) $. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n) $. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.Comment: A preliminary version of this paper appeared in the Proceedings of the Fourth International Computer Science Symposium in Russia, pages 356--367, 200

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience 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 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 present five examples where quantum finite automata (QFAs) outperform their classical counterparts. This may be useful as a relatively simple technique to introduce quantum computation concepts to computer scientists. We also describe a modern QFA model involving superoperators that is able to simulate all known QFA and classical finite automaton variants.

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 NQFA's, 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 QFA's. It is shown that these classes remain the same when the QFA's 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.