We will report on the properties and possible applications of a special class of maps facilitated by a set of one and two qubit quantum gates forming a quantum network applied iteratively on the initial state. One of the simplest possible quantum circuits, consisting of a CNOT gate, a Hadamard gate and a measurement on one of the outputs. It is known to lead to chaotic dynamics described in terms of complex rational maps. After studying the evolution of pure initial quantum states and reporting on their properties (like the formation of fractal structures (in the space of states)) we will examine how the ideal evolution is distorted in the presence of both coherent errors and incoherent initial noise, which are typical imperfections in current implementations of quantum computers. We show that the main features of the dynamics, and especially the fractal borders, are robust against certain noise, and that they will only be slightly distorted.