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We study the dynamics of quantum systems under classical and quantum noise,focusing on decoherence in qubit systems. Classical noise is described by arandom process leading to a stochastic temporal evolution of a closed quantumsystem, whereas quantum noise originates from the coupling of the microscopicquantum system to its macroscopic environment. We derive deterministic masterequations describing the average evolution of the quantum system underclassical continuous-time Markovian noise and two sets of master equationsunder quantum noise. Strikingly, these three equations of motion are shown tobe equivalent in the case of classical random telegraph noise and properquantum environments. Hence fully quantum-mechanical models within the Bornapproximation can be mapped to a quantum system under classical noise.Furthermore, we apply the derived equations together with pulse optimizationtechniques to achieve high-fidelity one-qubit operations under random telegraphnoise, and hence fight decoherence in these systems of great practicalinterest.Comment: 5 pages, 2 figures; converted to PRA format, added Fig. 2, corrected  typo

Equivalent qubit dynamics under classical and quantum noise