  In order to achieve the high-fidelity quantum control needed for a broad
range of quantum information technologies, reducing the effects of noise and
system inhomogeneities is an essential task. It is well known that a system can
be decoupled from noise or made insensitive to inhomogeneous dephasing
dynamically by using carefully designed pulse sequences based on square or
delta-function waveforms such as Hahn spin echo or CPMG. However, such ideal
pulses are often challenging to implement experimentally with high fidelity.
Here, we uncover a new geometrical framework for visualizing all possible
driving fields, which enables one to generate an unlimited number of smooth,
experimentally feasible pulses that perform dynamical decoupling or dynamically
corrected gates to arbitrarily high order. We demonstrate that this scheme can
significantly enhance the fidelity of single-qubit operations in the presence
of noise and when realistic limitations on pulse rise times and amplitudes are
taken into account.
