Single-qubit gates in two steps with rotation axes in a single plane

Any single-qubit unitary operation or quantum gate can be considered a rotation. Typical experimental implementations of single-qubit gates involve two or three fixed rotation axes, and up to three rotation steps. Here we show 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. The results are relevant for exchange-only logical qubits encoded in three-spin blocks, which are important for universal quantum computation in decoherence free subsystems and subspaces.