  We propose a variational principle to compute a quantum adiabatic
brachistochrone (QAB) for open systems. Using the notion of "adiabatic speed"
based on the energy gaps, we derive a Lagrangian associated to the functional
measuring the time spent to achieve adiabatic behavior, which in turn allows us
to perform the optimization. The QAB is illustrated for non-unitary dynamics of
STIRAP process, the Deutsch-Jozsa quantum computing algorithm and of a transmon
qutrit. A numerical protocol is devised, which allows to compute the QAB for
arbitrary quantum systems for which exact simulations can be afforded. We also
establish sufficient conditions for the equivalence between the Lagrangians,
and thus the QAB, of open and closed systems.
