Multi-step evolution and measurement control of finite-dimensional quantum systems

For finite-dimensional quantum systems, we propose a quantum control scheme based on a multi-step unitary evolution and quantum projective measurements. The objective is to design a control law to steer the system to a target eigenstate of the measurement operator in the least number of steps. Within each control step, unitary evolution and quantum projective measurement are performed in turn until the system reaches the target state. The control process can be modeled as a finite-state Markov chain with an absorbing state. We prove that the controlled system will converge to the target eigenstate with probability one after a finite number of control steps and find a minimal-step-number condition that would steer the system to the target eigenstate in the least number of steps.