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Quantum Finite State Machines (QFSM) are a well known model of computation that was originally formalized by Watrous [Wat95a, Wat95b, Wat97], Kondacs [KW97] and more generally Quantum Turing Machines (QTM) have been described by Bernstein [BV97]. In particular the 2-way QFSM have been shown to be more powerful than classical FSM [KW97]. Thus the interest in quantum computational models of automata and machines is not only theoretical but has also possible applications realization of future quantum computer and robotics controllers. In this chapter we present the evolutionary approach to the synthesis of QFSM’s specified by a quantum circuits. This approach was originally proposed by [LP09] and is possible on yet only theoretical basis. In particular this approach requires a selective qubit-initialization in a quantum register. In contrast the current methodology and approaches to practical Quantum Computation, the current practical realization of quantum computation always starts with the initialization of the whole quantum register and terminates by the measurement of either all of the qubits or by the measurement of a given subset of qubits. Moreover in general there is no reuse of any element of the quantum register. In this text we analyze in details what type of QFSM can be successfully synthesized. The evolutionary approach will evaluate the results based on both the correctness and the cost of the evolved machines. Multiple parameters such as type of error evaluation, synthesis constraints and evolutionary operators will be discussed when evaluating to the obtained results. In particular we show how to synthesize QFSMs as sequence detectors and illustrate their functionality both in the quantum world and in the classical (observable) world. The application of the synthesized quantum devices is illustrated by the analysis of recognized sequences. Finally, we provide analytic method for the used evolutionary approach and we describe the experimental protocol, and its heuristic improvements. We also discuss the results. In addition, we investigate the following aspects of the Evolutionary Quantum Logic Synthesis: • Quantum probabilistic FSM and Reversible FSM. • Hardware acceleration for the Fitness evaluation using CBLAS [cbl] and using CUBLAS [cud] (CUDA[cud] implemented Basic Linear Algebra Subprograms (BLAS)[cbl] subroutines).

Evolutionary Logic Synthesis of Quantum Finite State Machines for Sequence Detection