  We present an algebraic framework to study the time-optimal synthesis of
arbitrary unitaries in SU(2), when the control set is restricted to rotations
around two non-parallel axes in the Bloch sphere. Our method bypasses commonly
used control-theoretical techniques, and easily imposes necessary conditions on
time-optimal sequences. In a straightforward fashion, we prove that
time-optimal sequences are solely parametrized by three rotation angles and
derive general bounds on those angles as a function of the relative rotation
speed of each control and the angle between the axes. Results are substantially
different whether both clockwise and counterclockwise rotations about the given
axes are allowed, or only clockwise rotations. In the first case, we prove that
any finite time-optimal sequence is composed at most of five control
concatenations, while for the more restrictive case, we present scaling laws on
the maximum length of any finite time-optimal sequence. The bounds we find for
both cases are stricter than previously published ones and severely constrain
the structure of time-optimal sequences, allowing for an efficient numerical
search of the time-optimal solution. Our results can be used to find the
time-optimal evolution of qubit systems under the action of the considered
control set, and thus potentially increase the number of realizable unitaries
before decoherence.
