Stochastic Hybrid Systems for modelling and analysing Quantum Control Systems

Control in quantum systems is essential to develop quantum technologies. In general, advances in quantum technologies need a control theory to work properly. However, because of the peculiarities of the quantum mechanics, we... [ view full abstract ]

Control in quantum systems is essential to develop quantum technologies. In general, advances in quantum technologies need a control theory to work properly. However, because of the peculiarities of the quantum mechanics, we can not use classic control theory and it has been necessary to develop a quantum control theory. In this way, the theory of stochastic hybrid systems can be used. Stochastic hybrid hystems are dynamical systems that combine continuous evolution and instantaneous change and that include random effects. Quantum control systems are dynamical systems which evolve continuously and change instantaneously and randomly when a measurement is applied. Because of these common characteristics, we use the Stochastic Hybrid Systems framework to model and analyse Quantum Control Systems.

In this work we propose an stochastic hybrid system to model a two-level quantum control system with bounded uncertainties. The objective of this quantum control systems is to steer the system’s state into one of the eigenstates. For simplicity suppose that the state is on the other eigenstate. In this case, a control signal, which was designed previuosly, is used. After a certain period, we do a projective measurement. If the system’s state is in the wanted eigenstate, periodic projective measurements are implemented to hold this eigenstate. Else, the same control signal is applied followed by a projective measurement as before. In this model, projective measurements are used as control actions which manipulate the state. This model combines continuous evolution with random instantaneous changes and with switching of dynamics.

Then, we analyse the stability of this stochastic hybrid system for different types of uncertainties using Lyapunov functions. Moreover, we show examples and simulations of the stability of this model with the different types of uncertaintites.

As a result, we present an stochastic hybrid system able to model quantum control systems. With that, we can apply the results obtained in the stochastic hybryd systems framework to analyse quantum control systems. In this way, we obtain sufficient and necesaries condition for uniform global stability and uniform global asymptotic stability for different types of uncertainties.