Dephasing of qubits by transverse low-frequency noise

We analyze the dissipative dynamics of a two-level quantum system subject to low-frequency, e.g. $1/f$ noise, motivated by recent experiments with superconducting quantum circuits. We show that the effect of transverse linear coupling of the system to low-frequency noise is equivalent to that of quadratic longitudinal coupling. We further find the decay law of quantum coherent oscillations under the influence of both low- and high-frequency fluctuations, in particular, for the case of comparable rates of relaxation and pure dephasing.