  One of the most significant hurdles to be overcome on the path to practical
quantum information processors is dealing with quantum errors. Dynamical
decoupling is a particularly promising approach that complements conventional
quantum error correction by eliminating some correlated errors without the
overhead of additional qubits. In practice, the control pulses used for
decoupling are imperfect and thus introduce errors which can accumulate after
many pulses. These instrumental errors can destroy the quantum state. Here we
examine several dynamical decoupling sequences, and their concatenated
variants, using electron spin resonance of donor electron spins in a $^{28}$Si
crystal. All of the sequences cancel phase noise arising from slowly
fluctuating magnetic fields in our spectrometer, but only those sequences based
upon alternating $\pi$-rotations about the X- and Y-axes in the rotating frame
(XYXY sequences) demonstrate the ability to store an arbitrary quantum state.
By comparing the experimental results with a detailed theoretical analysis we
demonstrate that the superior performance of XYXY sequences arises from the
fact that they are self-correcting for the dominant instrumental pulse errors
in magnetic resonance experiments. We further find that concatenated sequences
perform better than the periodic variants, maintaining near 100% fidelities for
spin states even after several hundred control pulses. Intuitively, one would
expect the instrumental error to increase with the number of pulses in the
sequence but we show that the dominant first-order error does not increase when
concatenating the XYXY sequence.
