Minimization of a quantum automaton: The transducer
Ana María Guerra Martins · Physical Review A · 2008
A model of a quantum automaton (QA), a transducer, working with $n$ qubits is proposed. The quantum states of the automaton are represented by density operators. The linearity of the QA and of the partial trace operator, jointly with the properties of invariant subspaces and of the quotient spaces, are used to minimize the dimension and the cardinality of the state set of the QA.Two main results are obtained in this paper: (a) The conditions that the unitary evolution operators, representing the quantum gates of the QA, must obey in order to reduce its dimension and its state set. (b) An algorithm is developed to find the minimal QA and its complexity is computed. We show that the necessary and sufficient conditions for minimization depend uniquely in the structure of the unitary evolution operators, and they are independent of the initial quantum state of the QA.