Geometric formalism for constructing arbitrary single-qubit dynamically corrected gates

Implementing high-fidelity quantum control and reducing the effect of the coupling between a quantum system and environmental noise is a major challenge in developing quantum information technologies. In recent work, a geometrical pulse-shaping method was introduced to facilitate the design of time-optimized pulses that implement gates while suppressing quasistatic noise errors. This method works for resonant pulses that implement a subset of single-qubit gates corresponding to rotations of the qubit about an axis that is orthogonal to the noise fluctuation term in the qubit Hamiltonian. Here, we show that these earlier findings are a special case of a larger geometrical structure hidden within the time-dependent Schroedinger equation. In this framework, any noise-suppressing single-qubit gate corresponds to a closed three-dimensional space curve, where the driving fields that implement the robust gates can be extracted from the curvature and torsion of the space curves. This provides a systematic approach to obtain all possible driving fields that implement an arbitrary dynamically corrected gate in the presence of quasistatic noise. We show that similar geometrical structures exist for quantum systems of arbitrary Hilbert space dimension.