DOCSLIB.ORG

Simple Models of Quantum Finite Automata

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simple Models of Quantum Finite Automata Masaryk University Faculty of Informatics Simple models of quantum finite automata Bachelor’s Thesis Martin Frian Brno, Spring 2018 Masaryk University Faculty of Informatics Simple models of quantum finite automata Bachelor’s Thesis Martin Frian Brno, Spring 2018 This is where a copy of the official signed thesis assignment and a copy ofthe Statement of an Author is located in the printed version of the document. Declaration Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Martin Frian Advisor: prof. RNDr. Jozef Gruska DrSc. i Acknowledgements I would like to thank my supervisor prof. RNDr. Jozef Gruska, DrSc. for his guidance and valuable advice. iii Abstract We present in a uniform way several models of classical finite automata and quantum finite automata. We have studied five main models of quantum finite automata in detail. Three other models of quantum finite automata were also briefly presented. We have also presented basic results concerning the power of each model, closure properties, mutual relations, and results concerning decision problems. iv Keywords finite automata, quantum finite automata, classical state, quantum state, regular languages, acceptance probability, quantum information processing, quantum mechanics v Contents Introduction 1 1 Finite automata 3 2 Basic tools of QIP 9 3 Quantum finite automata 13 3.1 Measure-many one-way quantum finite automata ...... 13 3.2 Measure-once one-way qunatum finite automata ...... 17 3.3 Two-way quantum finite automata .............. 20 3.4 Enhanced quantum finite automata .............. 22 3.5 Quantum finite automata with open time evolution ..... 24 3.6 Other models of quantum finite automata .......... 25 4 Conlusion 27 Bibliography 29 vii Introduction In the classical world, each regular language can be recognized (de- scribed) by a finite automaton. Finite automata model can be seenas an abstract machine with an input tape, on which an input is written, each input symbol in a single cell, read by the reading head. Finite automaton can be in exactly one of a fixed finite set of states at any moment of discrete time. Depending on the input, this machine model can go from one state to another and this is called the transition. Ac- cording to the used transition function we have the two most basic models of finite automata, deterministic finite automata (DFA) and non-deterministic finite automata (NFA). In fact, despite the different definitions of these two models, their computational power issame. Trivially, every DFA is NFA too and it was proven that for each NFA there exists a DFA recognizing the same language. However such a DFA could have exponentially larger number of states than the NFA under recognition. Another computational model of finite automata are probabilistic finite automata (PFA). It is well known, that PFA have same computa- tional power as DFA at a certain definition of acceptance. In the quantum world, there exists also various types of computa- tional models of finite automata. Informally, quantum finite automata (QFA) are quantum analog of PFA. However, instead of using clas- sical states, they use quantum states. Depending on used quantum resources, QFA might be more powerful and they might have better time or space complexity than PFA or DFA. So far there are several models of QFA that use simple quantum tools but generate only a subset of regular languages. There is also a model of QFA that generates exactly the whole class of regular languages but use the whole range of quantum resources. Both of these models are not fully satisfactory. Hence we need a model of QFA, which recognises exactly the whole class of regular languages using as minimum of quantum tools as possible. Therefore the aim of this thesis is to provide an overview of the current well known main models of quantum finite automata as well as basic results concerning their power, mutual relations and decidability of basic decision problems. 1 This thesis is divided into four chapters. In the first chapter we present classical deterministic finite automata, their properties and several basic results concerning decidability of decision problems. We also show modifications of deterministic finite automata, namely non- deterministic finite automata, two-way deterministic finite automata and probabilistic finite autoamta. In the second chapter we provide a brief introduction to basic tools widely used in a construction of quan- tum finite automata. In the third chapter we present in detail current known models of quantum finite automata, namely measure-many one-way quantum finite automata, measure-once one-way quantum finite automata, two-way quantum finite automata, enhanced quan- tum finite automata and quantum finite automata with opentime evolution. We also briefly present another models of quantum finite automata. 2 1 Finite automata Deterministic finite automata (DFA) are basic models of classical in- formation processing devices. They consist of: 1. a finite memory of so-called states. One of the states is said to be an initial state. A subset of states is said to be a set of accepting states; 2. a set of input symbols called the alphabet; 3. a transition function, which specifies the action of the automaton during computations. For each state and each input symbol determines the next state of the automaton. Formally such an automaton is defined as follows. Definition 1. A deterministic finite automaton A (DFA) is a 5-tuple (S, S, d, s0, Sa), where ∙ S is a finite set of states ∙ S is an input alphabet ∙ d : S × S ! S is a transition function ∙ s0 2 S is an initial state ∙ Sa ⊆ S is a set of accepting states Processing of an input (string) w by a DFA A is informally defined as follows. The automaton starts processing w in the initial state and reads as its input the leftmost symbol of w. The automaton then pro- ceeds in discrete time steps as follows until it gets into an accepting state or finishes processing of the whole input string: for the current state and next symbol of the input string it comes to a new state of its memory that is determined by its transition function and moves to read next symbol of its input string. This process is formally defined via extended transition function. Definition 2. An Extended transition function is a function dˆ : S × S* ! S satisfying the following three conditions: 3 1. Finite automata ∙ d(s, e) = s ∙ dˆ(s, a) = d(s, a) ∙ dˆ(s, aw) = dˆ(d(s, a), w) for any a 2 S, w 2 S+ and s 2 S. A language accepted by a DFA is defined as follows. Definition 3. Let A = (S, S, d, s0, Sa) be a DFA, the language accepted or recognized by A, denote as L(A), is * L(A) = fw 2 S j dˆ(s0, w) 2 Sag One of the key results of finite automata theory is that the class of languages accepted by DFA is the class of regular languages. They are the smallest set of languages containing all finite languages and closed under the following operations: ∙ union, ∙ concatenation, ∙ iteration. DFA are of large importance for several reasons: ∙ It is easy to determine in polynomial time the smallest DFA equivalent to a given A. This smallest DFA is always unique. ∙ Main decision problems, namely the emptiness problem and the membership problem are decidable in polynomial time. The emptiness problem is in addition easily reduced to the reacha- bility problem on graphs. ∙ Equivalence problems, such as the equivalence of two states or two DFA are decidable in polynomial time too. ∙ The above model of DFA is very robust. Several of its natural modifications, presented bellow, accept again exactly the class of regular languages. Here are the main variants of the above model of DFA. 4 1. Finite automata Non-deterministic finite automata Informally, transitions in non-deterministic finite automaton are based on non-deterministic choosing of the next state. The transition function maps each pair state and input symbol to the set of potential next states. The automaton nondeterministically chooses one of them. A formal definition of NFA is as follows. Definition 4. A nondeterministic finite automaton (NFA) is a 5-tuple (S, S, d, s0, Sa), where meaning of all elements is same as in the Defini- tion 1 except the transition function that is defined as follows d : S × S !P(S)1. Again, the transition function could be extended in a similar way like in the Definition 2. dˆ : Q × S* !P(Q) satisfies the following conditions: ∙ dˆ(s, e) = fsg ∙ dˆ(s, wa) = S d(p, a) p2dˆ(s,w) for any s 2 S, a 2 S and w 2 S*. The language L(A) accepted by a NFA A is defined as follows * L(A) = fw 2 S j dˆ(s0, w) \ Sa 6= Æg Two-way finite automata Another generalization of DFA are two-way deterministic finite au- tomata (2DFA). The only difference between 2DFA and DFA is that in 2DFA the automaton could move in both directions while reading an input word or can stay on the same position during a transition. Now we provide a formal definition of 2DFA. Definition 5. A deterministic two-way finite automaton (2DFA) A is a 5-tuple A = (S, S, d, s0, Sa) where meaning of each element is same as in Definition 1, except the transition function defined as follows d : S × S ! S × f , #, !g 1. Here P(S) denotes the powerset of the set S 5 1.
Load more

Recommended publications
  • Quantum Algorithms 1
    Quantum Algorithms 1 Computer scientists usually find conventional expositions of quantum computation difficult, since a lot of new concepts have to be learned to- gether to make sense of it all. In this course, we take a different approach, starting from a point where every computer science student is comfortable, and introducing the new concepts one by one into the familiar framework. We hope it will be much easier this way. So let's start with deterministic finite automata! Chapter 1 Finite automata 1.1 Deterministic model You already know the stuff in Sipser's book. Note that the transitions of a DFA can be represented as 0-1 matrices, the states can be traced with vectors, and the execution on an input string boils down to multiplying this vector with the corresponding sequence of those matrices. 1.2 Probabilistic model We define real-time1 probabilistic finite automata (rtPFAs) by generalizing the matrix definition of rtDFAs to column stochastic transition matrices. Consider a transition graph representation for a moment. It is as if compu- tation flows through many arrows (associated with the same input symbol) parallelly in each step. Under this \zero-error" regime (exact computation) rtPFAs are identical to rtDFAs: Since none of these computational paths can arrive at an incorrect response at the end, tracing any single path is suffi- cient for seeing what the machine will do. So we convert each rtPFA matrix (which have to contain at least one nonzero entry in each column) to a rtDFA matrix with a 1 in just the position corresponding to the topmost nonzero entry of the rtPFA in that column.
    [Show full text]
  • Automata Modeling for Cognitive Interference in Users' Relevance
    Quantum Informatics for Cognitive, Social, and Semantic Processes: Papers from the AAAI Fall Symposium (FS-10-08) Automata Modeling for Cognitive Interference in Users’ Relevance Judgment Peng Zhang1, Dawei Song1, Yuexian Hou1,2, Jun Wang1, Peter Bruza3 1School of Computing, The Robert Gordon University, UK. 2School of Computer Sci. & Tec., Tianjin University, China. 3Queensland University of Technology, Australia Email: [email protected] Abstract This paper is a step further along the direction of inves- tigating the quantum-like interference in a central IR pro- Quantum theory has recently been employed to further ad- cess - user’s relevance judgment, where users are involved vance the theory of information retrieval (IR). A challenging to judge the relevance degree of documents with respect to research topic is to investigate the so called quantum-like in- a given query. In this process, users’ relevance judgment terference in users’ relevance judgment process, where users are involved to judge the relevance degree of each document for the current document is often interfered by the judgment with respect to a given query. In this process, users’ relevance for previously seen documents, as a result of cognitive in- judgment for the current document is often interfered by the terference. A typical scenario is that, after finding a more judgment for previous documents, due to the interference on relevant document, a user may lower the relevance degree users’ cognitive status. Research from cognitive science has of a previously judged document. For example, when a user demonstrated some initial evidence of quantum-like cogni- judges whether a document d0 about “the theory of relativ- tive interference in human decision making, which underpins ity” is relevant to the query “Albert Einstein”, the user might the user’s relevance judgment process.
    [Show full text]
  • Probabilistic and Quantum Finite Automata with Postselection
    Probabilistic and quantum finite automata with ⋆ ⋆⋆ postselection ’ Abuzer Yakaryılmaz and A. C. Cem Say Bo˘gazi¸ci University, Department of Computer Engineering, Bebek 34342 Istanbul,˙ Turkey abuzer,[email protected] June 26, 2018 Abstract. We prove that endowing a real-time probabilistic or quantum computer with the ability of postselection increases its computational power. For this purpose, we provide a new model of finite automata with postselection, and compare it with the model of L¯ace et al. We examine the related language classes, and also establish separations between the classical and quantum versions, and between the zero-error vs. bounded- error modes of recognition in this model. 1 Introduction The notion of postselection as a mode of computation was introduced by Aaronson [1]. Postselection is the (unrealistic) capability of discarding all branches of a computation in which a specific event does not occur, and focusing on the surviving branches for the final decision about the membership of the input string in the recognized language. Aaronson examined PostBQP, the class of languages recognized with bounded error by polynomial-time quantum computers with postselection, and showed it to be identical to the well-known classical complexity class PP. It is, arXiv:1102.0666v1 [cs.CC] 3 Feb 2011 however, still an open question whether postselection adds anything to the power of polynomial-time computation, since we do not even know whether P, the class of languages recognized by classical computers with zero error in polynomial time, equals PP or not. In this paper, we prove that postselection is useful for real-time computers with a constant space bound, that is, finite automata.
    [Show full text]
  • Quantum Finite Automata and Their Simulations
    Quantum Finite Automata and Their Simulations Gustaw Lippa Krzysztof Makie la Marcin Kuta [email protected] Department of Computer Science, AGH University of Science and Technology, Krak´ow, Poland Krak´owQuantum Informatics Seminar (KQIS) 13 October 2020 Outline Motivation Quantum finite automata Library for simulating quantum finite automata Existing libraries Java Formal Languages and Automata Package (JFLAP) Quirk Quantum++ Q# Qiskit ProjectQ Probabilistic automata Probabilistic Finite Automaton (PFA) Quantum automata Measure-Once Quantum Finite Automaton (MO-QFA) Measure-Many Quantum Finite Automaton (MM-QFA) General Quantum Finite Automaton (GQFA) Types of Finite Automata Classical automata Deterministic Finite Automaton (DFA) Nondeterministic Finite Automaton (NFA) Alternating Finite Automaton (AFA) Quantum automata Measure-Once Quantum Finite Automaton (MO-QFA) Measure-Many Quantum Finite Automaton (MM-QFA) General Quantum Finite Automaton (GQFA) Types of Finite Automata Classical automata Deterministic Finite Automaton (DFA) Nondeterministic Finite Automaton (NFA) Alternating Finite Automaton (AFA) Probabilistic automata Probabilistic Finite Automaton (PFA) Types of Finite Automata Classical automata Deterministic Finite Automaton (DFA) Nondeterministic Finite Automaton (NFA) Alternating Finite Automaton (AFA) Probabilistic automata Probabilistic Finite Automaton (PFA) Quantum automata Measure-Once Quantum Finite Automaton (MO-QFA) Measure-Many Quantum Finite Automaton (MM-QFA) General Quantum Finite Automaton (GQFA) Deterministic
    [Show full text]
  • Quantum Reversibility and a New Model of Quantum Automaton
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Quantum reversibility and a new model of quantum automaton Massimo Pica Ciamarra Dipartimento di Scienze Fisiche, Universit`a di Napoli \Federico II", Napoli, Italy. E-mail: [email protected] Abstract This article is an attempt to generalize the classical theory of reversible com- puting, principally developed by Bennet [IBM J. Res. Develop., 17(1973)] and by Fredkin and Toffoli [Internat. J. Theoret. Phys., 21(1982)], to the quantum case. This is a fundamental step towards the construction of a quantum computer be- cause a time efficient quantum computation is a reversible physical process. The paper is organized as follows. The first section reviews the classical theory of re- versible computing. In the second section it is showed that the designs used in the classical framework to decrease the consumption of space cannot be generalized to the quantum case; it is also suggested that quantum computing is generally more demanding of space than classical computing. In the last section a new model of fully quantum and reversible automaton is proposed. The computational power of this automaton is at least equal to that of classical automata. Some conclusion are drawn in the last section. 1 Reversible computing: a glance The classical theory of reversible computing has been analyzed extensively ([?, ?]). The main results are the following: 1. Every irreversible computation f : x f(x)canbe effectively enclosed in a reversible computation F :(x; 0) (f(x);x). →F is reversible because its input (x; 0) is uniquely determined by its output→ (f(x);x).
    [Show full text]
  • Arxiv:1111.3965V3 [Quant-Ph] 25 Jun 2012 Setting
    Quantum measurement occurrence is undecidable J. Eisert,1 M. P. Muller,¨ 2 and C. Gogolin1 1Qmio Group, Dahlem Center for Complex Quantum Systems, Freie Universitat¨ Berlin, 14195 Berlin, Germany 2Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we point out implications for measurement-based quantum computing and studies of quantum many-body models and suggest that a plethora of problems may indeed be undecidable. At the heart of the field of quantum information theory is the insight that the computational complexity of similar tasks in quantum and classical settings may be crucially different. Here we present an extreme example of this phenomenon: an operationally defined problem that is undecidable in the quan- tum setting but decidable in an even slightly more general classical analog. While the early focus in the field was on the assessment of tasks of quantum information processing, it has become increasingly clear that studies in computational complexity are also very fruitful when approaching problems FIG. 1. (Color online) The setting of sequential application of Stern- Gerlach-type devices considered here, gives rise to a tree of possible outside the realm of actual information processing, for exam- outcomes.
    [Show full text]
  • Lower Bounds for Quantum Computation and Communication
    Lower bounds for Quantum Computation and Communication by Ashwin V. Nayak B.Tech. (Indian Institute of Technology, Kanpur) 1995 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Umesh Vazirani, Chair Professor Alistair Sinclair Professor Raymond Chiao Fall 1999 The dissertation of Ashwin V. Nayak is approved: Chair Date Date Date University of California, Berkeley Fall 1999 Lower bounds for Quantum Computation and Communication Copyright c 1999 by Ashwin V. Nayak 1 Abstract Lower bounds for Quantum Computation and Communication by Ashwin V. Nayak Doctor of Philosophy in Computer Science University of California, Berkeley Professor Umesh Vazirani, Chair The description of the state of an n-bit quantum system requires 2n − 1 com- plex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing both exponential speed-up, and exponential savings in communication costs in solving certain problems using quantum computers. In this dis- sertation, we establish limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed. More specifically, we give tight bounds for random access codes, which allow us to encode classical information using quantum bits such that only a small portion of the encoded information may be recovered via a measure- ment. This also sheds light on the power of computing with a finite number of quantum bits|using these techniques, we show an exponential size lower bound for quantum finite automata for a problem which can be solved on a linear size classical automaton.
    [Show full text]
  • Quantum Computation and Learning
    I Quantum Computation and Learning Richard Bonner R¯usi¸nˇsFreivalds (Eds.) Third International Workshop, QCL 2002 Riga, Latvia Revised Proceedings II Copyright: R. Bonner, R. Freivalds, 2003 ISBN 91-88834-29-8 Printed by Arkitektkopia, V¨aster˚as, Sweden Distribution: Department of Mathematics and Physics, M¨alardalen University, V¨aster˚as, Sweden Preface In the background of any model of computation lies an assumption of a physical computational device, a physical system, the controlled dynamics of which is the computation. Classical models of computation tacitly assume an underlying classical and usually rather simple physical system. However, the miniaturization of electronic circuits on the one hand, and fundamental scientific questions on the other, have in recent years put this tacit assumption into question, giving birth to a rapidly growing field of quantum computation and information theory. Information may thus be stored in quantum physical systems and processed by controlling their dynamics. The present volume contributes to this new field. It is a collection of revised papers, originally presented at the international workshop QCL’2002 in Riga in May 2002. The papers are grouped into two separate sections, Quantum Com- putation, and, Learning, reflecting the fact that Quantum Learning - a goal of our investigations - is still a field in the making. Let us very briefly review the topics covered. The simplest model of quantum computation is the quantum finite automa- ton (QFA). There are several versions of this model, differentiated essentially by the mode of data access (1-way, 2-way, or hybrid), the way the quantum states are measured (measure-once or measure-many), the kind of quantum states admit- ted (pure or mixed), and the criteria of input acceptance (isolated or non-isolated cut-point).
    [Show full text]
  • Simulations of Quantum Finite Automata
    Simulations of Quantum Finite Automata Gustaw Lippa, Krzysztof Makie laand Marcin Kuta[0000−0002−5496−6287] Department of Computer Science, Faculty of Computer Science, Electronics and Telecommunications, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland, [email protected] Abstract. This paper presents a Python library to simulate different kinds of quantum finite automata on a classical computer. The library also provides tools for language generation and visual representation of simulation results. We have conducted experiments to measure time com- plexity of the simulation in function of automaton size, alphabet size and word length. Examples of library usage are also provided. Keywords: Quantum finite automata · Acceptance conditions · Cut- point · Automata simulation 1 Introduction Finite automata are real models of computers, which have only limited amount of memory. Finite automata are also interesting models themselves, due to their simplicity but at the same time rich structure and interesting properties [7]. Since introduction of finite automata in 1959 by Rabin and Scott [13], the theme have been quite well recognized and approached from various aspects. Connections with different domains, including algebra and logics, have also been established. The picture for theory of quantum automata and languages generated by them is less clear, and various important problems remain open [2, 11]. Task of quantum finite automata is to recognise quantum languages. Studying these languages is useful in establishing the expressivity and computational power of quantum machines in general. However, such devices are not yet available and simulators have to be used instead. Because of that, we developed a library writ- ten in Python, running on a classical computer and providing implementation of several types of quantum finite automata.
    [Show full text]
  • Models of Quantum Computation
    UNIVERSITY OF LATVIA OKSANA Š ČEGU ĻNAJA-DUBROVSKA MODELS OF QUANTUM COMPUTATION PHD THESIS Nozare: datorzin ātnes Apakšnozare: datorzin ātņu matem ātiskie pamati Promocijas darba vad ītājs: prof. , Dr.habil.math. RŪSI ŅŠ M ĀRTI ŅŠ FREIVALDS Rīga – 2010 1 ANOTĀCIJA Kvantu modelis ar p ēcatlasi tiek defin ēts Scott Aaronson darb ā. Beigu st āvok ļu kopa, kas parasti sast āv no akcept ējošiem un noraidošiem st āvok ļiem, tiek papildin āta ar parametru, kas nor āda, vai dotais beigu st āvoklis ietilpst atlases kop ā. M ērījumi tiek veikti tikai atlases kopas beigu st āvok ļos. Tiek ieviests papildus st āvoklis q+ , un ja visu p ēcatlases st āvok ļu amplit ūdas ir 0, tad q+ amplit ūda sa ņem v ērt ību 1. Pēcatlase ļauj p ētīt ne tikai kvantu, bet ar ī tradicion ālo algoritumu īpaš ības. Pētījuma m ērķis ir sal īdzin āt varb ūtisko un kvantu gal īgo p ēcatlases autom ātu klases un aprakst īt valodu klases, ko atpaz īst kvantu gal īgs autom āts ar p ēcatlasi. Pētījuma proces ā ieg ūti š ādi rezult āti: • Defin ēts kvantu gal īgā autom āta ar p ēcatlasi j ēdziens; • Aprakst īta valoda PALINDROMES, ko atpaz īst gal īgs kvantu autom āts ar pēcatlasi ar m ērījumu katr ā sol ī un gal īgs kvantu autom āts ar p ēcatlasi ar mērījumu beig ās; • Aprakst īta valoda, kuru nevar atpaz īt gal īgs kvantu autom āts ar p ēcatlasi ar mērījumu katr ā sol ī un gal īgs kvantu autom āts ar p ēcatlasi ar m ērījumu beig ās: L= {w | w ∈ *}1,0{ and there exist x, y,u, z such that w = 1yx = 1zu and x = z} Viens no promocijas darba uzdevumiem ir apl ūkot kvantu vaic ājošos algoritmus Bula funkciju r ēķ in āšanai.
    [Show full text]
  • Arxiv:1507.01988V2 [Cs.FL] 4 Jul 2018 Introduction 1 N Nci Brazil
    Automata and Quantum Computing Andris Ambainis and Abuzer Yakaryılmaz* University of Latvia, Faculty of Computing, Raina bulv. 19, R¯ıga, LV-1586, Latvia emails: [email protected] and [email protected] September 9, 2021 22 h 34 Abstract. Quantum computing is a new model of computation, based on quantum physics. Quan- tum computers can be exponentially faster than conventional computers for problems such as fac- toring. Besides full-scale quantum computers, more restricted models such as quantum versions of finite automata have been studied. In this paper, we survey various models of quantum finite automata and their properties. We also provide some open questions and new directions for re- searchers. 2010 Mathematics Subject Classification: 68Q10, 68Q12, 68Q15, 68Q19,68Q45 Key words: quantum finite automata, probabilistic finite automata, nondeterminism, bounded error, unbounded error, state complexity, decidability and undecidability, computational complexity Contents 1 Introduction 2 2 Mathematical background 3 3 Preliminaries 6 4 One-wayQFAs 7 4.1 Simulations ................................. 10 4.2 Succinctnessresults . 12 4.3 Bounded-error language recognition in restricted models ......... 13 4.4 Unbounded-error,nondeterminism,and alternation . ......... 14 4.5 Decidabilityandundecidabilityresults . ....... 15 4.5.1 Equivalenceandminimization.. 15 4.5.2 Emptiness problems and problems regarding isolated cutpoint. 16 arXiv:1507.01988v2 [cs.FL] 4 Jul 2018 4.5.3 Otherproblems..... .... .... ... .... .... .... 17 5 Two-way QFAs 17 5.1 2-wayQFAswithclassicalhead . 17 *A. Yakaryılmaz worked on the chapter when he was in Bo˘gazic¸i University in Turkey, University of Latvia, and LNCC in Brazil. 2 A. Ambainis, A. Yakaryılmaz 5.1.1 Bounded-errorlanguagerecognition. 18 5.1.2 Succinctnessresults.
    [Show full text]
  • Quantum Reversibility and a New Model of Quantum Automaton
    Quantum reversibility and a new model of quantum automaton Massimo Pica Ciamarra Dipartimento di Scienze Fisiche, Universit`adi Napoli “Federico II”, Napoli, Italy. E-mail: [email protected] Abstract This article is an attempt to generalize the classical theory of reversible com- puting, principally developed by Bennet [IBM J. Res. Develop., 17(1973)] and by Fredkin and Toffoli [Internat. J. Theoret. Phys., 21(1982)], to the quantum case. This is a fundamental step towards the construction of a quantum computer be- cause a time efficient quantum computation is a reversible physical process. The paper is organized as follows. The first section reviews the classical theory of re- versible computing. In the second section it is showed that the designs used in the classical framework to decrease the consumption of space cannot be generalized to the quantum case; it is also suggested that quantum computing is generally more demanding of space than classical computing. In the last section a new model of fully quantum and reversible automaton is proposed. The computational power of this automaton is at least equal to that of classical automata. Some conclusion are drawn in the last section. arXiv:quant-ph/0102104v1 21 Feb 2001 1 Reversible computing: a glance The classical theory of reversible computing has been analyzed extensively ([2, 3]). The main results are the following: 1. Every irreversible computation f : x → f(x) can be effectively enclosed in a reversible computation F : (x, 0) → (f(x), x). F is reversible because its input (x, 0) is uniquely determined by its output (f(x), x).
    [Show full text]