Efficiency of dynamical decoupling sequences in presence of pulse errors

For a generic dynamical decoupling sequence employing a single-axis control, we study its efficiency in the presence of small errors in direction of the controlling-pulses. In the case that the corresponding ideal dynamical-decoupling sequence produces good results, the impact of the errors is found to scale as $\xi^2$, with negligible first-order effect, where $\xi$ is the dispersion of the random errors. This analytical prediction is numerically tested in a model, in which the environment is modeled by one qubit coupled to a quantum kicked rotator in chaotic motion. In this model, with periodic pulses applied to the qubit in the environment, it is shown numerically that Uhrig dynamical decoupling is not necessarily better than the bang-bang control.