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Quantum vs. classical pushdown automata in exact computation

Yumiko Murakami

Published Mar 24, 2004• 0 citations
OpenAlex

Abstract

Since some quantum algorithms were introduced by Shor [1] or by Grover [2], much attention has been given to the field of quantum computation.The quantum computation is so useful for solving certain problems, however, the classical computation is more powerful in some cases.Thus, it is significant to compare the ability of quantum computation with the ability of classical counterpart, based on simple computation models, like automata.In this paper, we focus on the quantum pushdown automata, QPAs, which was defined by Golovkins in 2000 [4].It is shown that the class of languages recognized by QPAs contains the class of languages recognized by finite automata.However, no one knows the relationships between the cognitive ablity of QPAs and the classical counterparts.We make a proposition in this paper, the QPAs that deterministically solve a certain problem, which cannot be solved by deterministic pushdown automata.

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Since some quantum algorithms were introduced by Shor [1] or by Grover [2], much attention has been given to the field of quantum computation.The quantum computation is so useful for solving certain problems, however, the classical computation is more powerful in some cases.Thus, it is significant to compare the ability of quantum computation with the ability of classical counterpart, based on simple computation models, like automata.In this paper, we focus on the quantum pushdown automata, QPAs, which was defined by Golovkins in 2000 [4].It is shown that the class of languages recognized by QPAs contains the class of languages recognized by finite automata.However, no one knows the relationships between the cognitive ablity of QPAs and the classical counterparts.We make a proposition in this paper, the QPAs that deterministically solve a certain problem, which cannot be solved by deterministic pushdown automata.
Even though quantum computation is useful for solving certain problems, classical computation is more powerful in some cases. Thus, it is significant to compare the abilities of quantum computation and its classical counterpart, based on such a simple computation model as automata. In this paper we focus on the quantum pushdown automata which were defined by Golovkins in 2000, who showed that the class of languages recognized by quantum pushdown automata properly contains the class of languages recognized by finite automata. However, no one knows the entire relationship between the recognitive abilities of quantum and classical pushdown automata. As a part, we show a proposition that quantum pushdown automata can deterministically solve a certain problem that cannot be solved by any deterministic pushdown 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).
Quantum finite automata, as well as quantum pushdown automata (QPA) were first introduced by C. Moore and J. P. Crutchfield. In this paper we introduce the notion of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of Kondacs and Watrous. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, not recognizable by deterministic pushdown automata.
In general, quantum computation models are expected to be more powerful than classical counterparts. However, sometimes this is not the case.It is known that there exists some regular language which one-way quantum finite automata and one-way quantum counter automata cannot recognize. This is due to the restriction of reversibility which quantum models must satisfy. Thus, it is an interesting question: what kinds of quantum models suffer from/overcome the restriction? To tackle this problem, we focus on (empty-stack acceptance) one-way quantum pushdown automata, and show that there exists a regular language which one-way quantum pushdown automata cannot recognize. This implies adding a stack to one-way finite automata cannot overcome the restriction of reversibility if we adopt empty-stack acceptance as the acceptance mode.
One important question for quantum computing is whether a computational gap exists between models that are allowed to use quantum effects and models that are not. Several types of quantum computation models have been proposed, including quantum finite automata and quantum pushdown automata (with a quantum pushdown stack). It has been shown that some quantum computation models are more powerful than their classical counterparts and others are not since quantum computation models are required to obey such restrictions as reversible state transitions. In this paper, we investigate the power of quantum pushdown automata whose stacks are assumed to be implemented as classical devices, and show that they are strictly more powerful than their classical counterparts under the perfect-soundness condition, where perfect-soundness means that an automaton never accepts a word that is not in the language. That is, we show that our model can simulate any probabilistic pushdown automata and also show that there is a non-context-free language which quantum pushdown automata with classical stack operations can recognize with perfect soundness.
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.