  Implementing high-fidelity quantum control and reducing the effect of the
coupling between a quantum system and its environment is a major challenge in
developing quantum information technologies. Here, we show that there exists a
geometrical structure hidden within the time-dependent Schr\"odinger equation
that provides a simple way to view the entire solution space of pulses that
suppress noise errors in a system's evolution. In this framework, any
single-qubit gate that is robust against quasistatic noise to first order
corresponds to a closed three-dimensional space curve, where the driving fields
that implement the robust gate can be read off from the curvature and torsion
of the space curve. Gates that are robust to second order are in one-to-one
correspondence with closed curves whose projections onto three mutually
orthogonal planes each enclose a vanishing net area. We use this formalism to
derive new examples of dynamically corrected gates generated from smooth
pulses. We also show how it can be employed to analyze the noise-cancellation
properties of pulses generated from numerical algorithms such as GRAPE. A
similar geometrical framework exists for quantum systems of arbitrary Hilbert
space dimension.
