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Quantum Automata Formalism

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These are general questions related to complexity of quantum algorithms, combinational and sequential

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Models of quantum sequential circuits 1.Quantum automata 2.Quantum state machines 3.Quantum Turing Machines 4.Quantum Robots of Benioff 5.Quantum Cellular Automata (not quantum dot based).

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A new formalism for classical (deterministic) automata

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Input state 1 Input state 2 Output state 1 Output state 2 This means that external classical computer has to change the quantum circuit when a new input in the string comes Observe that this matrix is not permutative and not unitary

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A formalism for classical non-deterministic automata

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Nondeterminism for b Observe that this matrix is not permutative and not unitary

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bb There are two paths from state 1 to state 2, which have labels sequence bb Using matrices like these we can analyze if certain transitions in graphs exist and how many of them exist. This is used in finding the languages accepted be the automata

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A FORMALISM FOR CLASSICAL PROBABILISTIC AUTOMATA

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PROBABILISTIC AUTOMATA

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Languages accepted by probabilistic automata

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A FORMALISM FOR QUANTUMAUTOMATA

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Now unitary matrices Quantum Finite Automata = QFA

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Unitary matrix ket bra Probability that an automaton accepts a string

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Languages accepted by deterministic automata Review the following: 1. the concept of Rabin-Scott automaton and language accepted by it. 2.Review the concept of regular expression 3.Show a link between regular expression and language accepted by an automaton. 4.Language generated by an automaton. 5.Regular languages

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Languages accepted by probabilistic automata Unitary matrices used here are only a subset of all matrices

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Model of Quantum Automaton Quantum automaton is programmed from deterministic standard automaton. It is more similar to FPGA than normal model of computing like in a processor. Quantum Automaton CLASSICAL AUTOMATON One pulse for one elementary rotation in one qubit 1.Machine here has a program that generates pulses that program QA. 2.This is like a memory in FPGA that stores information about LUT and connections Finite memory Infinite memory

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unitary matrix Quantum Automaton described by a unitary matrix

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CLASSICAL TURING MACHINES

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Classical Turing Machines

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polynomial Model of calculation of a standard Turing Machine

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Example of Turing Machine tape head Finite State Machine 1.Move left, 2.move right, 3.stop, 4.write a symbol. 5.Is the symbol in current cell Xi? The source of infiniteness is the tape This machine has a finite memory, this is standard automaton. Automaton control the head

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Non-Polynomial Non-Polynomial, these are tough problems in real life

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Bounded-error probabilistic polynomial (BPP) In computational complexity theory, bounded-error probabilistic polynomial time (BPP) is the class of decision problems that are:computational complexity theory decision problems 1.solvable by a probabilistic Turing machineprobabilistic Turing machine 2.in polynomial time,polynomial time 3.with an error probability of at most 1/3 for all instances.probability

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Bounded-error probabilistic polynomial Informally, a problem is in BPP if there is an algorithm for it that has the following properties: 1.It is allowed to flip coins and make random decisions 2.It is guaranteed to run in polynomial time 3.On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer, whether the answer is YES or NO. BPP = Bounded-error Probabilistic Polynomial A complexity class

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Bounded-error probabilistic polynomial

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QUANTUM TURING MACHINES TURING MACHINES

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A sum of two complex numbers can be a zero

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BQP bounded error quantum polynomial time 1.In computational complexity theory BQP (bounded error quantum polynomial time) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances.computational complexity theorydecision problemspolynomial time 2.It is the quantum analogue of the complexity class BPP.BPP 3.In other words, there is an algorithm for a quantum computer (a quantum algorithm) that solves the decision problem with high probability and is guaranteed to run in polynomial time.algorithmquantum algorithm 4. On any given run of the algorithm, it has a probability of at most 1/3 that it will give the wrong answer.

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BQP (cont) 1.Similarly to other "bounded error" probabilistic classes the choice of 1/3 in the definition is arbitrary. 2.We can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff bound.Chernoff bound 3.Detailed analysis shows that the complexity class is – unchanged by allowing error as high as 1/2 − n −c on the one hand, – or requiring error as small as 2 −nc on the other hand, where c is any positive constant, and n is the length of input.

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Sources:

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Used in 2011.

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