This paper proposed a quantum analogue of classical queue automata by using the definition of the quantum Turing machine and quantum finite-state automata. However, quantum automata equipped with storage medium of a stack has been considered, but the concept of quantum queue automata has not been introduced so far. The classical Turing machines can be simulated by classical queue automata. Motivated by the efficiency of the quantum Turing machine and nature of classical queue automata, we have introduced the notion of quantum queue automata using unitary criteria. Our contributions are as follows. We have also introduced a generalization of real-time deterministic queue automata, the real-time quantum queue automata which work in real-time i.e. the input head can move towards the right direction only and takes exactly one step per input symbol. We have shown that real-time quantum queue automata is more superior than its real-time classical variants by using quantum transitions. We have proved the existence of the language that can be recognized by real-time quantum queue automata and cannot be recognized by real-time deterministic (reversible) queue automata. Further, we have shown that there is a language that can be recognized by real-time quantum queue automata but not by real-time non-deterministic queue automata.
This paper proposed a quantum analogue of classical queue automata by using the definition of the quantum Turing machine and quantum finite-state automata. However, quantum automata equipped with storage medium of a stack has been considered, but the concept of quantum queue automata has not been introduced so far. The classical Turing machines can be simulated by classical queue automata. Motivated by the efficiency of the quantum Turing machine and nature of classical queue automata, we have introduced the notion of quantum queue automata using unitary criteria. Our contributions are as follows. We have also introduced a generalization of real-time deterministic queue automata, the real-time quantum queue automata which work in real-time i.e. the input head can move towards the right direction only and takes exactly one step per input symbol. We have shown that real-time quantum queue automata is more superior than its real-time classical variants by using quantum transitions. We have proved the existence of the language that can be recognized by real-time quantum queue automata and cannot be recognized by real-time deterministic (reversible) queue automata. Further, we have shown that there is a language that can be recognized by real-time quantum queue automata but not by real-time non-deterministic queue automata.
Quantum finite automata were introduced by C.Moore and J.P.Crutchfield in [MC 97] and by A.Kondacs and J.Watrous in [KW 97]. This notion is not a generalization of the deterministic finite automata, but rather a generalization of deterministic reversible (permutation) automata. In [AF 98] A.Ambainis and R.Freivalds raised the question what kind of probabilistic automata can be viewed as a special case of quantum finite automata. To answer that question and study relationship between quantum finite automata and probabilistic finite automata, we introduce a notion of probabilistic reversible automata (PRA, or doubly stochastic automata). We give the necessary condition for a language to be recognized by PRA. We find that there is a strong relationship between different possible models of PRA and corresponding models of quantum finite automata. In these thesis we regard quantum automata, probabilistic reversible and deterministic reversible automata as reversible automata. At least two non-equivalent definitions of language recognition are used for one-way reversible finite automata in various papers. Both of these definitions have their own advantages and disadvantages, summarized in our classification of one-way reversible automata. Further, we introduce a notion of quantum finite multitape automata and prove that there is a language recognized by a quantum finite multitape automaton but not by deterministic finite multitape automata. Additionally we discover unexpected probabilistic automata recognizing complicated languages. Finally, we revise the concept of quantum pushdown automata (QPA), first introduced by C.Moore and J.P.Cruthchfield in [MC 97]. We give the definition of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of [KW 97]. It is established that the unitary criteria of QPA are not equivalent to the corresponding unitary criteria of quantum Turing machines [BV 97]. We show that QPA can recognize every regular language. We also present some simple non-context-free languages recognized by QPA (hence not recognizable by deterministic and nondeterministic pushdown automata).
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.
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 automata theory, the quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to the QFAs augmented with counters or stacks. Moreover, to our knowledge, there is no result related to QFAs having more than one input head. In this paper, we focus on such generalizations of QFAs whose input head(s) operate(s) in one-way or realtime mode and present many superiority of them to their classical counterparts. Furthermore, we propose some open problems and conjectures in order to investigate the power of quantumness better. We also give some new results on classical computation.
This paper introduces a variant of two-way quantum finite automata named two-way multihead quantum finite automata. A two-way quantum finite automaton is more powerful than classical two-way finite automata. However, the generalizations of two-way quantum finite automata have not been defined so far as compared to one-way quantum finite automata model. We have investigated the newly introduced automata from two aspects: the language recognition capability and its comparison with classical and quantum counterparts. It has been proved that a language which cannot be recognized by any one-way and multi-letter quantum finite automata can be recognized by two-way quantum finite automata. Further, it has been shown that a language which cannot be recognized by two-way quantum finite automata can be recognized by two-way multihead quantum finite automata with two heads. Furthermore, it has been investigated that quantum variant of two-way deterministic multihead finite automata takes less number of heads to recognize a language containing of all words whose length is a prime number.
The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.