Implementation of single-qubit gates with two rotations around axes in a plane

Typical experimental implementations of single-qubit gates involve two or three fixed rotation axes, and up to three rotation steps. Reducing the number of steps from three to two therefore represents a huge savings. In this work, we prove that, if the rotation axes can be tuned arbitrarily in a fixed plane, then two rotation steps are sufficient for implementing a single-qubit gate, and one rotation step is sufficient for implementing a state transformation. As concrete examples, we demonstrate two-step single-qubit gate implementations in two different physical qubit systems: (i) a transmon superconducting qubit coupled to an external microwave drive, such as a transmission line; (ii) a quantum-dot based exchange-only qubit encoded in a three-spin block. These results provide a significant speedup for many common gate implementations, such as Rabi oscillations with phase control.