Two-way Quantum Finite Automata Research Articles

Learning quantum finite automata with queries

AbstractLearning finite automata (termed as model learning) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity (two-way QFA are more powerful than classical finite automata in computation ability as well). As a different problem in quantum learning theory and quantum machine learning, in this paper, our purpose is to initiate the study of learning QFA with queries (naturally it may be termed as quantum model learning), and the main results are regarding learning two basic one-way QFA (1QFA): (1) we propose a learning algorithm for measure-once 1QFA (MO-1QFA) with query complexity of polynomial time and (2) we propose a learning algorithm for measure-many 1QFA (MM-1QFA) with query complexity of polynomial time, as well.

A Quantum Finite Automata Approach to Modeling the Chemical Reactions

In recent years, the modeling interest has increased significantly from the molecular level to the atomic and quantum scale. The field of computational chemistry plays a significant role in designing computational models for the operation and simulation of systems ranging from atoms and molecules to industrial-scale processes. It is influenced by a tremendous increase in computing power and the efficiency of algorithms. The representation of chemical reactions using classical automata theory in thermodynamic terms had a great influence on computer science. The study of chemical information processing with quantum computational models is a natural goal. In this paper, we have modeled chemical reactions using two-way quantum finite automata, which are halted in linear time. Additionally, classical pushdown automata can be designed for such chemical reactions with multiple stacks. It has been proven that computational versatility can be increased by combining chemical accept/reject signatures and quantum automata models.

On the power of two-way multihead quantum finite automata

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.

Modeling of RNA secondary structures using two-way quantum finite automata

Quantum finite automata (QFA) play a crucial role in quantum information processing theory. The representation of ribonucleic acid (RNA) and deoxyribonucleic acid (DNA) structures using theory of automata had a great influence in the computer science. Investigation of the relationship between QFA and RNA structures is a natural goal. Two-way quantum finite automata (2QFA) is more dominant than its classical model in language recognition. Motivated by the concept of quantum finite automata, we have modeled RNA secondary structure loops such as, internal loop and double helix loop using two-way quantum finite automata. The major benefit of this approach is that these sequences can be parsed in linear time. In contrast, two-way deterministic finite automata (2DFA) cannot represent RNA secondary structure loops and two-way probabilistic finite automata (2PFA) can parse these sequences in exponential time. To the best of authors knowledge this is the first attempt to represent RNA secondary structure loops using two-way quantum finite automata.

Exact Affine Counter Automata

We introduce an affine generalization of counter automata, and analyze their ability as well as affine finite automata. Our contributions are as follows. We show that there is a language that can be recognized by exact realtime affine counter automata but by neither 1-way deterministic pushdown automata nor realtime deterministic k-counter automata. We also show that a certain promise problem, which is conjectured not to be solved by two-way quantum finite automata in polynomial time, can be solved by Las Vegas affine finite automata. Lastly, we show that how a counter helps for affine finite automata by showing that the language MANYTWINS, which is conjectured not to be recognized by affine, quantum or classical finite state models in polynomial time, can be recognized by affine counter automata with one-sided bounded-error in realtime.

Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata

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 is verifier. 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 Stockmeyer, 1990) and one-way quantum finite automata as well as two-way quantum finite automata (Nishimura and Yamakami, 2009). IP in which verifiers use 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 Lknapsack, 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 cannot be recognized by any AM(2PFA), and that the NP-complete language Lknapsack can also be recognized by a QAM(2QCFA) working only on quantum pure states using unitary operators.

Efficient probability amplification in two-way quantum finite automata

In classical computation, one only needs to sequence O ( log 1 ϵ ) identical copies of a given probabilistic automaton with one-sided error p < 1 to run on the same input in order to obtain a two-way machine with error bound ϵ . For two-way quantum finite automata (2qfa’s), this straightforward approach does not yield efficient results; the number of machine copies required to reduce the error to ϵ can be as high as ( 1 ϵ ) 2 . In their celebrated proof that 2qfa’s can recognize the non-regular language L = { a n b n ∣ n > 0 } , Kondacs and Watrous use a different probability amplification method, which yields machines with O ( ( 1 ϵ ) 2 ) states, and with runtime O ( 1 ϵ | w | ) , where w is the input string. In this paper, we examine significantly more efficient techniques of probability amplification. One of our methods produces machines which decide L in O ( | w | ) time (i.e. the running time does not depend on the error bound) and which have O ( ( 1 ϵ ) 2 c ) states for any given constant c > 1 . Other methods, yielding machines whose state complexities are polylogarithmic in 1 ϵ , including one which halts in o ( log ( 1 ϵ ) | w | ) time, are also presented.

Some algebraic properties of measure-once two-way quantum finite automata

Quantum finite automata (QFA) can be divided into four kinds depend upon the head-directions and the measure times. They are measure-once one way QFA (MO-1QFA) introduced by Moore and Crutchfield (Theor Comput Sci 237: 275---306, 2000); measure-many one way QFA (MM-1QFA) and measure-many two-way QFA (MM-2QFA) introduced by Kondacs and Watrous (Proceedings of the 38th IEEE annual symposium on 433 foundations of computer science, 66---75, 1997); and measure-once two-way QFA (MO-2QFA) which were not given until now. The purpose of this work is mainly to discuss one kind of QFA, which is called MO-2QFA for brief. First of all, the definition of MO-2QFA is given and the conditions for preserving unitary properties are shown. Then, we analysis the basic algebraic properties of the class of languages which can be recognized by MO-2QFA, such as the union, intersection, complement and reversal operations. As well, we consider the catenation operation on the class of quantum languages recognized by MO-2QFA is closed in the generalized conditions.