  Qubits, the quantum mechanical bits required for quantum computing, must
retain their fragile quantum states over long periods of time. In many types of
electron spin qubits, the primary source of decoherence is the interaction
between the electron spins and nuclear spins of the host lattice. For electrons
in gate defined GaAs quantum dots, previous spin echo measurements have
revealed coherence times of about 1 $\mu$s at low magnetic fields below 100 mT.
Here, we show that coherence in such devices can actually survive to much
longer times, and provide a detailed understanding of the measured nuclear spin
induced decoherence. At fields above a few hundred millitesla, the coherence
time measured using a single-pulse spin echo extends to 30 $\mu$s. At lower
magnetic fields, the echo first collapses, but then revives at later times
given by the period of the relative Larmor precession of different nuclear
species. This behavior was recently predicted, and as we show can be
quantitatively accounted for by a semi-classical model for the electron spin
dynamics in the presence of a nuclear spin bath. Using a multiple-pulse
Carr-Purcell-Meiboom-Gill echo sequence, the decoherence time can be extended
to more than 200 $\mu$s, which represents an improvement by two orders of
magnitude compared to previous measurements. This demonstration of effective
methods to mitigate nuclear spin induced decoherence puts the quantum error
correction threshold within reach.
