  We study the decoherence of a qubit weakly coupled to frustrated spin baths.
We focus on spin-baths described by the classical Ising spin glass and the
quantum random transverse Ising model which are known to have complex
thermodynamic phase diagrams as a function of an external magnetic field and
temperature. Using a combination of numerical and analytical methods, we show
that for baths initally in thermal equilibrium, the resulting decoherence is
highly sensitive to the nature of the coupling to the environment and is
qualitatively different in different parts of the phase diagram. We find an
unexpected strong non-Markovian decay of the coherence when the random
transverse Ising model bath is prepared in an initial state characterized by a
finite temperature paramagnet. This is contrary to the usual case of
exponential decay (Markovian) expected for spin baths in finite temperature
paramagnetic phases, thereby illustrating the importance of the underlying
non-trivial dynamics of interacting quantum spinbaths.
