In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by Díaz-Caro and Yakaryılmaz [6] referred as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which allows us to characterize the class of exclusive stochastic languages. Then, we initiate a study on the succintness of affine automata. In particular, we show that an infinite family of unary regular languages can be recognized by 2-state affine automata but the state numbers of quantum and probabilistic automata cannot be bounded. Finally, we present the characterization of all (regular) unary languages recognized by two-state affine automata.

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SUMMARYIn this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step toward this objective, the following question is being addressed: Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps? This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general S U(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric S U(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution. key words: quantum computation, random walks, quantum walks, asymptotic approximation

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