  In this paper, we study time-optimal control problems related to system of
two coupled qubits where the time scales involved in performing unitary
transformations on each qubit are significantly different. In particular, we
address the case where unitary transformations produced by evolutions of the
coupling take much longer time as compared to the time required to produce
unitary transformations on the first qubit but much shorter time as compared to
the time to produce unitary transformations on the second qubit. We present a
canonical decomposition of SU(4) in terms of the subgroup SU(2)xSU(2)xU(1),
which is natural in understanding the time-optimal control problem of such a
coupled qubit system with significantly different time scales. A typical
setting involves dynamics of a coupled electron-nuclear spin system in pulsed
electron paramagnetic resonance experiments at high fields. Using the proposed
canonical decomposition, we give time-optimal control algorithms to synthesize
various unitary transformations of interest in coherent spectroscopy and
quantum information processing.
