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Quantum Computing 8 QUANTUM COMPUTING 8. Jozef Gruska Faculty of Informatics Brno Czech Republik October 27, 2020 Quantum computing 8, 2020 8. QUANTUM FINITE AUTOMATA as models of small memory quantum processors Jozef Gruska October 27, 2020 1 Quantum computing 8, 2020 WHY to STUDY QUANTUM PROCESSORS WITH SMALL MEMORY? - I. There are three main reasons why it is of interest and importance to investigate models of quantum processors that use little of quantum resources. and importance. 1. Experiences with classical information processing strongly suggest that investigation of the models of processors with finite memory and working in real-time brings a huge variety of fundamental theoretical results for information processing and, at the same time, provides a variety of methods and tools to design such processors. 2. Two decades of the research in quantum information processing start to make clear that not very powerful, with hundreds of qubits, quantum processors, but processors using only little of quantum resources, are to be seen currently as the goal that could be both feasible and worth to make { because theoretical results have already demonstrated that they can, in some cases, bring significant increase of the information processing power. 3. Processors combining huge classical and small quantum resources should therefore be seen as the next goal in quantum information processing once the current concentration of the research on few qubits storage, few particles entanglement and few unitary operations circuits start to rich fully satisfactory outcomes. Jozef Gruska October 27, 2020 2 Quantum computing 8, 2020 MODELS OF QUANTUM AUTOMATA Models of quantum automata For most of the main classical models of automata there are also their quantum versions. For example for finite automata, Turing machines and quantum cellular automata (QCA). Models of quantum automata are used: • To get an insight into the power of different quantum computing models and modes, using language/automata theoretic methods. • To discover the simplest models of computation at which one can demonstrate large (or huge) difference in the power of quantum versus classical models. • To develop quantum automata (networks, algorithms) design methodologies. • To explore mutual relations between different quantum computation models and modes. • To discover, in a transparent and elegant form, limitations of quantum computations and communications. Jozef Gruska October 27, 2020 3 Quantum computing 8, 2020 WHY to STUDY QUANTUM PROCESSORS WITH SMALL MEMORY? - II. Using such models one can also get deeper insides about: • How much more power, concerning computation, can bring quantum (randomness) resources comparing with classical (randomness) resources? • How much more power can have computation in linear time comparing with with that in real time? • How much more power can bring computation with mixed states comparing that with pure states? • When and how much more power can bring addition of very little of quantum resources? Jozef Gruska October 27, 2020 4 Quantum computing 8, 2020 MAIN MODELS of CLASSICAL PROCESSORS Main classical models of processors are: • Finite automata (deterministic, non-deterministic, probabilistic, ultrametric,...., one-way, two-way,...; one-tape, multi-tape,....) • Turing machines (with one or more tapes, with one or more heads, with one or more dimensional tapes - deterministic, non-deterministic, probabilistic,...) - models of universal processors • Uniform classes of circuits - models of universal processors • Cellular automata (one-, two-, three- and more dimensional){ models of universal processors. Of importance, especially for an understanding of the power and development of methods (of programming) for very powerful real processors are also the following models: • RAM - Random access machines • PRAM - Parallel and shared memory random access machines Jozef Gruska October 27, 2020 5 Quantum computing 8, 2020 MAIN MODELS of QUANTUM AUTOMATA 1. QUANTUM FINITE AUTOMATA (QFA) QFA are considered to be the simplest model of quantum processors, with “finite” quantum memory, that models well the most basic mode of quantum computing | a quantum action is performed on each classical input. 2. QUANTUM (one-tape and many-tape) TURING MACHINES (QTM) QTM are used to explore, at the most general level of sequential computation, the potential and limitations of quantum computing. Using this model the main computational complexity classes are defined. QTM are a main quantum abstraction of human computational processes. 3. QUANTUM CELLULAR AUTOMATA (QCA) QCA are used to model and to explore, on every general and basic level of parallel computation, the potential and limitations of quantum computing. QCA are a very basic quantum abstraction of computation by nature. Main classical modes of computation: deterministic, nondeterministic and randomized. Jozef Gruska October 27, 2020 6 Quantum computing 8, 2020 BASIC MODELS of CLASSICAL FINITE AUTOMATA • Deterministic (one-way) finite automata. • Deterministic two-way finite automata. • Nondeterministic finite automata (one-way or two-way) • Probabilistic (randomized) versions of finite automata ab c x y z # ab c x y z $ σ q control 1FAq control 2FAq control (a) unit (b) unit (c) unit q q 0.5 0.5 q’ q’’ q’ q’’ NFA PFA 0.2 0.80.3 0.7 (d) q1q2 q 3 q 4 q1q2 q 3 q 4 (e) Figure 1: Models of finite automata Jozef Gruska October 27, 2020 7 Quantum computing 8, 2020 FROM MODELS of CLASSICAL AUTOMATA to MODELS of QUANTUM AUTOMATA For the most of the main classical models of automata there are also their quantum versions. Models of quantum automata are used: • To get a deeper insight into the power of different quantum computing models and modes, using language/automata theoretic methods. • To discover the simplest models of computation at which one can clearly demonstrate large (or huge) difference in the power of quantum versus classical models. • To develop quantum automata (networks, algorithms) design methodologies. • To explore mutual relations between different quantum computation models and modes. • To discover, in a transparent and elegant form, limitations of quantum computations and communications. Jozef Gruska October 27, 2020 8 Quantum computing 8, 2020 MAIN MODES of COMPUTATION of FINITE AUTOMATA Basic modes: Deterministic automata (DFA): In each step next configuration is uniquely (deterministically) determined. An input is accepted if it makes the automaton to reach an accepted state. Non-deterministic automata (NFA): In each step one of possible configurations is taken. An input is accepted if it can make the automaton to reach an accepted state. Probabilistic automata (PFA): In each step each possible configuration has a probability assigned. An input is accepted if it makes automaton to reach an accepting state with a certain probability - several modes of that are considered. Ultrametric automata (UFA): In each step each possible transition has a p-adic number assigned. Acceptance of an input is considered as in probabilistic case. Quantum automata (QFA): In each step each possible configuration has a complex number, probability amplitude, assigned and automaton can be at any step in a superposition of configurations. An input is accepted if it makes automaton to reach a configuration of accepting states with a certain probability. Jozef Gruska October 27, 2020 9 Quantum computing 8, 2020 MAIN PROPERTIES of CLASSICAL FINITE AUTOMATA • The concept of classical finite automata is very robust - all main variants of the basic model have the same recognition power. • Informally, classical finite automata can do exactly that what can be done in real-time - in time needed to read input - and with finite (input length independent) memory. • Formally, classical finite automata can recognize exactly the class of regular languages - the class of languages containing finite languages and closed under operations of union, concatenation and iteration. • For each regular language L there is a unique minimal finite deterministic automaton accepting L - minimization problem is therefore well defined. • There is an efficient algorithm to find the minimal deterministic finite automaton with. • All major decision problems for finite classical automata are decidable. • To study various classes of languages accepted by classical finite automata a variety of rich algebraic methods can be used to get deep insights. Jozef Gruska October 27, 2020 10 Quantum computing 8, 2020 MAIN PROBLEMS to EXPLORE for MODELS of QUANTUM FINITE AUTOMATA. • Recognition (processing) power of various models of quantum automata - properties and characterization of the classes of languages accepted by different models of quantum automata. • What kind of acceptance probabilities are achievable? • Succinctness relation to the corresponding models of quantum automata - how much smaller can quantum automata be for doing the same task as classical automata of the similar type. • Decidability of basic decision problems - for example of the equivalence problem. • Decidability of basic decision problems - for example of the equivalence problem. • Development and complexity study of methods to construct minimal quantum automata for the case of various models of quantum automata. Jozef Gruska October 27, 2020 11 Quantum computing 8, 2020 MAIN DIVISIONS of QUANTUM MODELS of FINITE AUTOMATA There are three main ways to divide models of quantum finite automata: 1. According to head moves and power: • all heads have always to read the same symbol and have always move in one direction • all heads can
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