  One of the simplest possible quantum circuits, consisting of a CNOT gate, a
Hadamard gate and a measurement on one of the outputs is known to lead to
chaotic dynamics when applied iteratively on an ensemble of equally prepared
qubits. The evolution of pure initial quantum states is characterized by a
fractal (in the space of states), formed by the border of different convergence
regions. We examine how the ideal evolution is distorted in the presence of
both coherent error and incoherent initial noise, which are typical
imperfections in current implementations of quantum computers. It is known that
under the influence of initial noise only, the fractal is preserved, moreover,
its dimension remains constant below a critical noise level. We systematically
analyze the effect of coherent Hadamard gate errors by determining fixed points
and cycles of the evolution. We combine analytic and numerical methods to
explore to what extent the dynamics is altered by coherent errors in the
presence of preparation noise as well. We show that the main features of the
dynamics, and especially the fractal borders, are robust against the discussed
noise, they will only be slightly distorted. We identify a range of error
parameters, for which the characteristic properties of the dynamics are not
significantly altered. Hence, our results allow to identify reliable regimes of
operation of iterative protocols.
