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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). Five of the presented papers fall into the category of general QFA theory, which compares different QFA with one another, and with their classical, de- terministic or probabilistic, counterparts. Golovkins and Kravtsev propose a new class of probabilistic automata with doubly stochastic transition matrices; this class is close, in a well-defined sense, to the QFA. Bonner, Freivalds and Rasˇsˇcevskis find a language for the recognition of which the number of states required by a 1-way measure-many QFA is exponentially smaller than that re- quired by a classical deterministic automaton; Midrijanis, on the other hand, presents another language, for which the relation is roughly the opposite. Ozols finds a language not recognized by deterministic automata, but recognized by 1-way measure-many QFA with non-isolated cut-point. Dubrovsky and Scegul- naja compare QFA with pure states and QFA with mixed states with respect to Boolean function computation and language recognition; they present a lan- guage recognized by the QFA with mixed states with a better probability than achievable by admitting pure states only. Four authors consider QFA in a more specific context: V. Kravcevs, Lace and Kuzmenko - for Boolean function computation, and Tervits - for undecidability proofs. Kravcevs and Lace compare the performance of quantum and probabilis- tic query automata (decision trees) on specific functions, Kuzmenko constructs quantum automata for the computation of certain DNF forms in a single query, IV and Tervits constructs a QFA which recognizes a language of key importance to an indecidability problem. Two authors, Kuzmenko (Query Automata Minimization) and Luchko, con- sider the problem of evolving a quantum automaton for a given task by means of genetic algorithms. There are three papers on classical learning. Tervits considers finite stan- dardizability, proving that this type of learning is not much less powerful than Gold’s identification in the limit. Kaulis studies the synthesis of logic formu- lae from finite examples, obtaining necessary and sufficient conditions in many- sorted first-order predicate logic with equality. Greizina and Grundmane take up a special problem of Learning from Zero Information. Finally, quantum learning is considered in the two papers by Bonner and Freivalds. One of the papers discusses limited memory learning by 1-way QFA in the ‘identification in the limit’ model, showing that it is more powerful than its classical counterpart. The other paper reviews recent results on quantum learning. We thank all contributors, referees, and our Program Committee. We grate- fully acknowledge financial sponsorship from several institutions, specified below and in the papers, and in particular the initial support of the Swedish Institute for our ML2000 project, which stared it all. Richard Bonner R¯usi¸nˇsFreivalds Organization The International Workshop QCL’2002, on Quantum Computation and Learn- ing, was organized jointly by the Institute of Mathematics and Computer Sci- ence, University of Latvia, Riga, and the Department of Mathematics and Physics, M¨alardalen University, V¨aster˚as, Sweden. It was the third of annual QCL work- shops, started within a joint research project ML2000, under financial spon- sorship of the Swedish Institute. Working papers from the previous workshops, QCL’1999 and QCL’2000, are available online at http://www.ima.mdh.se/personal/rbr/courses/QCL.htm. Program Committee Farid Ablayev (Kazan, Russia) Richard Bonner (V¨aster˚as, Sweden) Cristian S. Calude (Auckland, New Zealand) Alexandre Dikovsky (Nantes, France) Rusins Freivalds (Riga, Latvia) Alexander Letichevsky (Kiiv, Ukraine) Samuel J. Lomonaco, Jr. (Baltimore, U.S.A.) Arun Sharma (Sydney, Australia) Mars Valiev (Moscow, Russia) Research and Workshop Sponsors The European Commission The Latvian Council of Science The Swedish Institute A/S Dati Grupa The University of Latvia The Department of Mathematics and Physics, M¨alardalen University VI Table of Contents Quantum Computation Probabilistic Reversibility and its Relation to Quantum Automata ...... 1 Marats Golovkins and Maksim Kravtsev On the Size of Finite Probabilistic and Quantum Automata ............ 19 Richard Bonner, R¯usi¸nˇsFreivalds and Zigm¯ars Rasˇsˇcevskis Computation of Boolean Functions by Probabilistic and Quantum Decision Trees ................................................... 26 Vasilijs Kravcevs Some Examples to Show the Advantages of Probabilistic and Quantum Decision Trees ................................................... 32 Lelde L¯ace A Quantum Single-Query Automaton ............................... 40 Dmitry Kuzmenko Deterministic Finite Automata and 1-way Measure-Many Quantum Finite Automata with Non-isolated Cutpoint ........................ 47 Raitis Ozols The Complexity of Probabilistic versus Quantum Finite Automata ...... 51 Gatis Midrij¯anis Probabilistic and Quantum Automata for Undecidability Proofs ........ 57 Gints Tervits Quantum Automata with Mixed States: Function Computation and Language Recognition ............................................. 62 Andrej Dubrovsky and Oksana Scegulnaja Query Automata Minimization ..................................... 67 Dmitry Kuzmenko Generation of Quantum Finite Automata by Genetic Programming...... 71 Alexey Luchko Learning Inductive Inference of Logic Formulas ............................... 74 D¯avis K¯ulis VIII Quantum Learning by Finite Automata .............................. 85 Richard Bonner and R¯usi¸nˇsFreivalds Capabilities of Finite Standardization ............................... 97 Gints Tervits A Survey of Quantum Learning ..................................... 106 Richard Bonner and R¯usi¸nˇsFreivalds Learning from Zero Information..................................... 120 Madara Greizi¸na and L¯ıga Grundmane Probabilistic Reversibility and its Relation to Quantum Automata Marats Golovkins ? and Maksim Kravtsev?? Institute of Mathematics and Computer Science University of Latvia, Riga, Latvia [email protected], [email protected] Abstract. To study the relationship between quantum finite automata and probabilistic finite automata, we introduce a notion of probabilistic reversible automata (PRA), or doubly stochastic automata. We find that there is a strong relationship between different possible models of PRA and corresponding models of quantum finite automata. We also propose a classification of reversible finite 1-way automata. 1 Introduction Two models of probabilistic reversible automata (PRA) are examined in this paper, namely, 1-way PRA and 1.5-way PRA. Presently, we outline the notions applicable to both models in a quasi-formal way, including a general notion of probabilistic reversibility; formal definitions are provided in further sections. Introductory notions If not specified otherwise, we denote by Σ an input alpha- bet of an automaton, Σ∗ then being the set of its finite words. Every input word in Σ∗ is assumed enclosed within end-marker symbols # and $; it is therefore convenient to also introduce a working alphabet Γ = Σ #, $ . By Q we nor- mally understand the set of states of an automaton. By∪ {L we} understand the complement of a language L Σ∗. For an input word ω in #Σ∗$, we denote by ⊂ ω the number of symbols in ω, and by [ω]i, i = 1, 2,..., the i-th consecutive symbol| | of ω, excluding end-markers. Presented with a word ω on its input tape, a probabilistic automaton reads ω one symbol at a time, each time changing state; the next symbol and next state S being determined only in probabilistic sense. In particular, we write q q0, −→ S Σ∗, if there is a positive probability of getting from state q to state q0 by ⊂ reading ω S. We define the configuration of an automaton as a triple c = νqν0 ∈ h i
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