  A stochastic procedure is developed which allows one to express Pontryagin's
maximum principle for dissipative quantum system solely in terms of stochastic
wave functions. Time-optimal controls can be efficiently computed without
computing the density matrix. Specifically, the proper dynamical update rules
are presented for the stochastic costate variables introduced by Pontryagin's
maximum principle and restrictions on the form of the terminal cost function
are discussed. The proposed procedure is confirmed by comparing the results to
those obtained from optimal control on Lindbladian dynamics. Numerically, the
proposed formalism becomes time and memory efficient for large systems, and it
can be generalized to describe non-Markovian dynamics.
