Efficient numerical optimization using quantum trajectories

The wave-function Monte-Carlo method, also referred to as the use of "quantum trajectories", allows for the efficient simulation of open systems by independently tracking the evolution of many pure-state "trajectories". Numerical optimal control theory is a versatile tool for the design of control fields that steer a quantum system towards some goal, e.g. the creation of a highly entangled state. Here we show that Krotov's method of optimal control can be modified in a simple way to be fully parallel in the pure states. This provides a highly efficient method for finding optimal control protocols for open quantum systems and networks with typically large Hilbert spaces. We apply this method to the problem of generating entangled states in a network. We show that due to the existence of a dark-state subspace, nearly-optimal control protocols can be found by using only a single pure-state in the optimization, further increasing the efficiency.