ℋ₂ Model Reduction for Linear Quantum Systems

In this paper, an ℋ₂ model reduction method for linear quantum systems is presented, which can obtain a physically realizable model with a reduced order for closely approximating the original system. The model reduction problem is formulated as an optimization problem, which minimizes the ℋ₂ norm of the difference between the transfer functions of the original and reduced models. Different from classical model reduction, physical realizability conditions for guaranteeing that the reduced-order system is also a quantum system should be taken as non-convex constraints in the optimization. Based on the special properties of quantum systems, we derive necessary conditions for ℋ₂ optimality. Moreover, we employ a matrix inequality approach to transform nonlinear inequality constraints into readily solvable linear matrix inequalities (LMIs) and nonlinear equality constraints, so that it can be solved by a lifting variables approach. We emphasize that different from existing works, which only introduces a criterion to evaluate the performance after model reduction, we guide our method to obtain an optimal reduced model with respect to the ℋ₂ norm. Finally, a comparison with the results in the existing works validates the efficacy of the proposed method.