Quantum computing defines the use of quantum mechanics in the computation process. Given that Moore's law is about to reach its physical limits, it is natural to search for new technologies and computation models. The need for mechanisms and verification processes for infinite quantum computation will draw the attention of the research community. Potential applications of the proposed automata concern the verification of quantum systems and the design of appropriate quantum circuits. In the present work we define a quantum version of ω-automata, the periodic quantum automata that are able to efficiently recognize ω-languages of the form (amb)ω, where m is a finite number. We present our definitions through examples and figures. Even if our definition is simple, yet it has several interesting and useful properties. For example, they seem to be space efficient for periodic ω-languages of the form (amb)ω.
Quantum computing defines the use of quantum mechanics in the computation process. Given that Moore's law is about to reach its physical limits, it is natural to search for new technologies and computation models. The need for mechanisms and verification processes for infinite quantum computation will draw the attention of the research community. Potential applications of the proposed automata concern the verification of quantum systems and the design of appropriate quantum circuits. In the present work we define a quantum version of ω-automata, the periodic quantum automata that are able to efficiently recognize ω-languages of the form (amb)ω, where m is a finite number. We present our definitions through examples and figures. Even if our definition is simple, yet it has several interesting and useful properties. For example, they seem to be space efficient for periodic ω-languages of the form (amb)ω.
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 the contemporary era we dwell in, every now and then, new technologies are developed by master brains such as researchers and scientists. This also paves the way to discover replacements to classical computers. One such progression among the great inventions is the essence of quantum computing. They are brought to light for the purpose of solving problems and scenarios solved by classical computers but in a more effective way and in the least span of time. The study of quantum computing focuses on finding new ways to compute by using quantum physics phenomena. A qubit, in contrast to a typical computer bit, can be either 0 or 1, or it can be a superposition of both 0 and 1. Quantum theory commemorates with automata theory, thereby formulating quantum finite automata. Quantum finite automaton, or basically a deterministic finite automaton, is one of the simplest models of computation, which is an input-driven model, which takes an input, and according to the input, we just read the machine, and it changes its state. This proves that any quantum finite automata are completely capable of recognizing languages like Tally with given cut point too. Such languages recognized by a cut point, by quantum finite automata, are called stochastic languages. It further collaborates with the chemical industry with its qubit power over molecular collide, interactive models, and collaborative string processing using nonregular languages.
Game theory and its quantum extension apply in numerous fields that affect people’s social, political, and economical life. Physical limits imposed by the current technology used in computing architectures (e.g., circuit size) give rise to the need for novel mechanisms, such as quantum inspired computation. Elements from quantum computation and mechanics combined with game-theoretic aspects of computing could open new pathways towards the future technological era. This paper associates dominant strategies of repeated quantum games with quantum automata that recognize infinite periodic inputs. As a reference, we used the PQ-PENNY quantum game where the quantum strategy outplays the choice of pure or mixed strategy with probability 1 and therefore the associated quantum automaton accepts with probability 1. We also propose a novel game played on the evolution of an automaton, where players’ actions and strategies are also associated with periodic quantum automata.
The study of quantum finite automata (QFA's) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as MODp={aj∣j≡0modp}, where p is a prime number. In this paper, we present a modified MCQFA algorithm for the language MODp, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using less basis gates compared to the implementation of the original algorithm given in the literature.
A quantum finite-state automaton (QFA) is a theoretical model designed to simulate the evolution of a quantum system with finite memory in response to sequential input strings. We define the language of a QFA as the set of strings that lead the QFA to an accepting state when processed from its initial state. QFAs exemplify how quantum computing can achieve greater efficiency compared to classical computing. While being one of the simplest quantum models, QFAs are still notably challenging to construct from scratch due to the preliminary knowledge of quantum mechanics required for superimposing unitary constraints on the automata. Furthermore, even when QFAs are correctly assembled, the limitations of a current quantum computer may cause fluctuations in the simulation results depending on how an assembled QFA is translated into a quantum circuit. We present a framework that provides a simple and intuitive way to build QFAs and maximize the simulation accuracy. Our framework relies on two methods: First, it offers a predefined construction for foundational types of QFAs that recognize special languages MOD and EQU. They play a role of basic building blocks for more complex QFAs. In other words, one can obtain more complex QFAs from these foundational automata using standard language operations. Second, we improve the simulation accuracy by converting these QFAs into quantum circuits such that the resulting circuits perform well on noisy quantum computers. Our framework is available at https://github.com/sybaik1/qfa-toolkit.
The computational model of Quantum Finite Automata has been introduced by multiple authors (e.g. [38, 44]) with some variations in definition. The objective of this thesis is to understand what class of languages can be recognized by these different variations, and how many states are required. We begin by showing that we can use algebraic automata theory to characterize the language recognition power of QFAs. Algebraic automata theory associates to each language a canonical syntactic monoid, and the algebraic structure of this monoid becomes a meaningful parameter in describing language classes. We show that the class of languages recognized by Latvian QFAs [3] corresponds exactly to boolean combinations of languages recognized by Brodsky and Pippenger's QFA model [20], which correspond exactly to those languages whose syntactic monoid is in the class BG. Known results give us a decision procedure for testing membership in this language class. We also use algebraic automata theory to give nearly tight upper and lower bounds on the class of languages recognized by Brodsky and Pippenger's QFAs. We then extend a number of lower bound techniques known for Kondacs and Watrous' 1-way QFA model to Nayak's Generalized QFA. Both of these models are related in that they are permitted to halt before reading the entire input, allowing them to recognize certain languages whose syntactic monoid lies outside of BG. Finally, we investigate the question of QFA succinctness. It is known that QFAs can recognize some languages using exponentially fewer states compared to deterministic finite automata. We extend results from [16] to show that the word problem over abelian groups has this property. We also give example of interesting noncommutative languages with this property.