Potential of Quantum Finite Automata with Exact Acceptance

1998

We study 1-way quantum finite automata (QFAs). First, we compare them with their classical counterparts. We show that, if an automaton is required to give the correct answer with a large probability (greater than 7/9), then any 1-way QFA can be simulated by a 1-way reversible automaton. However, quantum automata giving the correct answer with smaller probabilities are more powerful than reversible automata.

Equality and disjointness are two of the most studied problems in communication complexity. They have been studied for both classical and also quantum communication and for various models and modes of communication. Buhrman et al. [6] proved that the exact quantum communication complexity for a promise version of the equality problem is O(log n) while the classical deterministic communication complexity is n + 1 for two-way communication, which was the first impressively large (exponential) gap between quantum and classical (deterministic and probabilistic) communication complexity. If an error is tolerated, both quantum and probabilistic communication complexities for equality are O(log n). However, even if an error is tolerated, the gaps between quantum (probabilistic) and deterministic complexity are not larger than quadratic for the disjointness problem [2, 4, 16, 18, 24, 25]. It is therefore interesting to ask whether there are some promise versions of the disjointness problem for which bigger gaps can be shown. We give a positive answer to such a question. Namely, we prove that there exists an exponential gap between quantum (even probabilistic) communication complexity and classical deterministic communication complexity of some specific versions of the disjointness problem. Klauck [16] proved, for any language, that the state complexity of exact quantum/classical finite automata, which is a general model of one-way quantum finite automata, is not less than the state complexity of an equivalent one-way deterministic finite automata (1DFA). In this paper we show, using a communication complexity result, that situation may be different for some promise problems. Namely, we show for certain promise problem that the gap between the state complexity of exact one-way quantum finite automata and 1DFA can be exponential.

Information and Computation, 2012

Several types of automata, such as probabilistic and quantum automata, require to work with real and complex numbers. For such automata the acceptance of an input is quantified with a probability. There are plenty of results in the literature addressing the complexity of checking the equivalence of these automata, that is, checking whether two automata accept all inputs with the same probability. On the other hand, the critical problem of finding the minimal automata equivalent to a given one has been left open [C. Moore, J.P. Crutchfield, Quantum automata and quantum grammars, Theoret. Comput. Sci. 237 275-306, see p. 304, Problem 5]. In this work, we reduce the minimization problem of probabilistic and quantum automata to finding a solution of a system of algebraic polynomial (in)equations. An EXPSPACE upper bound on the complexity of the minimization problem is derived by applying Renegar's algorithm. More specifically, we show that the state minimization of probabilistic automata, measure-once quantum automata, measure-many quantum automata, measure-once generalized quantum automata, and measure-many generalized quantum automata is decidable and in EXPSPACE. Finally, we also solve an open problem concerning minimal covering of stochastic sequential machines [A. Paz, Introduction to Probabilistic Automata, Academic Press, New York, 1971, p. 43].

Quantum Inf. Process., 2021

Two of the most well-known quantum algorithms, those introduced by Deutsch–Jozsa and Bernstein–Vazirani, can solve promise problems with just one function query, showing an oracular separation with deterministic classical algorithms. In this work, we generalise those methods to study a family of quantum algorithms that can, with just one query, exactly solve promise problems stated over Boolean functions. We also show that these problems can be naturally ordered, inducing a partially ordered set of promise problems. We study the properties of such a poset, showing that the Deutsch–Jozsa and Bernstein–Vazirani problems are, in a certain sense, extremal problems in it, determining some of its automorphisms and proving that it is connected. We also prove that, for the problems in the poset, the corresponding classical query complexities can take any value between 1 and $$2^{n-1}+1$$ 2 n - 1 + 1 .

Information and Computation

Interactive proof systems (IP) are very powerful-languages they can accept form exactly PSPACE. They represent also one of the very fundamental concepts of theoretical computing and a model of computation by interactions. One of the key players in IP are verifiers. In the original model of IP whose power is that of PSPACE, the only restriction on verifiers is that they work in randomized polynomial time. Because of such key importance of IP, it is of large interest to find out how powerful will IP be when verifiers are more restricted. So far this was explored for the case that verifiers are two-way probabilistic finite automata (Dwork and Stockmayer, 1990) and one-way quantum finite automata as well as two-way quantum finite automata (Nishimura and Yamakami, 2009). IP in which verifiers uses public randomization is called Arthur-Merlin proof systems (AM). AM with verifiers modeled by Turing Machines augmented with a fixed-size quantum register (qAM) were studied also by Yakaryilmaz (2012). He proved, for example, that an NP-complete language L knapsack , representing the 0-1 knapsack problem, can be recognized by a qAM whose verifier is a two-way finite automaton working on quantum mixed states using superoperators. In this paper we explore the power of AM for the case that verifiers are two-way finite automata with quantum and classical states (2QCFA)-introduced by Ambainis and Watrous in 2002-and the communications are classical. It is of interest to consider AM with such "semi-quantum" verifiers because they use only limited quantum resources. Our main result is that such Quantum Arthur-Merlin proof systems (QAM(2QCFA)) with polynomial expected running time are more powerful than the models in which the verifiers are two-way probabilistic finite automata (AM(2PFA)) with polynomial expected running time. Moreover, we prove that there is a language which can be recognized by an exponential expected running time QAM(2QCFA), but can not be recognized by any AM(2PFA), and that the NP-complete language L knapsack can also be recognized by a QAM(2QCFA) working only on quantum pure states using unitary operators.

2007

Lecture Notes in Computer Science, 2014

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. Some parts of the material are based on the lectures given by the second author during his visits to Kazan Federal University, Ural Federal University, and Bogaziçi University in 2013. Yakaryılmaz was partially supported by CAPES, ERC Advanced Grant MQC, and FP7 FET project QALGO.

2007

Abstract Lots of efforts in the last decades have been done to prove or disprove whether the set of polynomially bounded problems is equal to the set of polynomially verifiable problems. This paper will present an overview of the current beliefs of quantum complexity theorists and discussion detail the impacts these beliefs may have on the future of the field. We introduce a new form of Turing machine based on the concepts of quantum mechanics and investigate the domain of this novel definition for computation.

acadlib.lv

The thesis assembles research on two models of automata-probabilistic reversible (PRA) that appear very similar to 1-way quantum finite automata (1-QFA) and quantum one-way one counter automata (Q1CA), that is the most restricted model of non-finite space quantum automata. The objective of the research is to describe classes of languages recognizable by these models and compare related quantum and probabilistic automata. We propose the model of probabilistic reversible automata. We study both one-way PRA with classical (1-C-PRA) and decide and halt (1-DH-PRA) acceptance. We show recognition of general class of languages L n = a * 1 a * 2. .. a * n with probability 1 − ε. We show whether the classes of languages they recognize are closed under boolean operations and describe general class of languages not recognizable by these automata in terms of "forbidden constructions" for the minimal deterministic automaton of the language. We also consider "weak" reversibility as equivalent definition for 1-way automata and show the difference from ordinary reversibility in 1.5-way case. We propose the general notion of quantum one-way one counter automata(Q1CA). We describe well-formedness conditions for the Q1CA that ensure unitarity of its evolution. A special kind of Q1CA, called simple, that satisfies the well-formedness conditions is introduced. We show recognition of several non context free languages by Q1CA. We show that there is a language that can be recognized by quantum one-way one counter automaton, but not by the probabilistic one counter automaton. 10. Quantum Computation and Learning. 1st International Workshop. Riga, Latvia, September 11-13, 1999. Presentation "Quantum One Counter Automata". I thank my co-authors Richard Bonner, Rūsiņš Freivalds, Marats Golovkins, Vasilijs Kravcevs, who significantly contributed to this research. I thank Arnolds Ķ ikusts and Andrey Dubrovski for useful discussions. Especially I thank my supervisor Prof. Rūsiņš Freivalds, whose ideas, support and positive attitude was one of the key factors for me to pursue and complete the research. CONTENTS 3.6.2 1.5-way Probabilistic Reversible Automata. .. .. .. 4 Quantum one way 1 counter automata 4.1 Definition of Q1CA .

2019

We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post's correspondence problem.