Exact solution for the dynamical decoupling of a qubit with telegraph noise

We study the dissipative dynamics of a qubit that is afflicted by classical random telegraph noise and it is subject to dynamical decoupling. We derive exact formulas for the qubit dynamics at arbitrary working points in the limit of infinitely strong control pulses (bang-bang control) and we investigate in great detail the efficiency of the dynamical decoupling techniques both for Gaussian and non-Gaussian (slow) noise at qubit pure dephasing and at optimal point. We demonstrate that control sequences can be successfully implemented as diagnostic tools to infer spectral proprieties of a few fluctuators interacting with the qubit. The analysis is extended in order to include the effect of noise in the pulses and we give upper bounds on the noise levels that can be tolerated in the pulses while still achieving efficient dynamical decoupling performance.