Time-optimal state transfer for an open qubit

Finding minimal time and establishing the structure of the corresponding optimal controls which can transfer a given initial state of a quantum system into a given target state is a key problem of quantum control. In this work, this problem is solved for a basic component of various quantum technology processes -- a qubit interacting with the environment and experiencing an arbitrary time-dependent coherent driving. We rigorously derive both upper and lower estimates for the minimal steering time. Surprisingly, we discover that the optimal controls have a very special form -- they consist of two impulses, at the beginning and at the end of the control period, which can be assisted by a smooth time-dependent control in between. Moreover, an important for practical applications explicit almost optimal state transfer protocol is provided which only consists of four impulses and gives an almost optimal time of motion. The results can be directly applied to a variety of experimental situations for estimation of the ultimate limits of state control for quantum technologies.