State transition diagrams for a universal quantum gate set

A quantum Turing machine (QTM) is an abstract model of computing that serves as the quantum counterpart of a classical Turing machine. Closely related to the probabilistic Turing machine, a QTM utilises quantum effects such as superposition, entanglement and unitary evolution. Despite its historical role as a framework for devising quantum algorithms and the significance of its classical counterpart in theoretical computing, very little attention has been paid in literature to the state transition of QTM’s for basic quantum gates. In this paper, we construct the state transition diagrams for a set of elementary quantum gates that consists of the Hadamard, CNOT, and T gates, providing a universal basis for fault tolerant quantum computation. We verify the necessary conditions to ensure that the designed state transition diagrams comply with the postulates of quantum mechanics, verifying their well formedness.