Construction of Arbitrary Robust One-Qubit Operations Using Planar Geometry

We show how to construct an arbitrary robust one-qubit unitary operation with a control Hamiltonian of $A_x(t){\sigma }_x+A_y(t){\sigma }_y$, where ${\sigma }_i$ is a Pauli matrix and $A_i(t)$ is piecewise constant. Our method, based on planar geometry, admits a simple and intuitive interpretation. Furthermore, the total execution time and the number of elementary gates of the obtained sequence are comparable to those of the shortest known concatenated composite pulses.