We study the problem of robust performance of quantum systems under structured uncertainties. A specific feature of closed (Hamiltonian) quantum systems is that their poles lie on the imaginary axis and that neither a coherent controller nor physically relevant structured uncertainties can alter this situation. This changes for open systems where decoherence ensures asymptotic stability and creates a unique landscape of pure performance robustness, with the distinctive feature that closed-loop stability is secured by the underlying physics and needs not be enforced. This stability, however, is often detrimental to quantum-enhanced performance, and additive perturbations of the Hamiltonian give rise to dynamic generators that are nonlinear in the perturbed parameters, invalidating classical paradigms to assess robustness to structured perturbations such as singular value analysis. This problem is addressed using a fixed-point iteration approach to determine a maximum perturbation strength δmaxthat ensures that the transfer function remains bounded, ║Tδ║−1for δmax.
This paper analyzes how the weight function affects the properties of convergence: where and how fast the estimate converges. Stabilization control of quantum systems has been getting much attention as research in quantum information technology progresses, and of course, it is necessary to consider uncertainties in practice. In order to deal with a large uncertain quantum system, we recently proposed an adaptive parameter tuning algorithm based on the study of robust stabilization of
N -level quantum systems by Liang, Amini, and Mason (SIAM J. Control Optim., 59(1), 669–692, 2021). In this paper, we briefly review our proposed method and analyze its convergence performance of the estimated parameter. The convergence analysis shows that certain conditions on the weight function are necessary for proper tuning, and the asymptotic convergence rate depends on only the choice of the function.
This technical note extends applications of the quantum small gain and Popov methods from existing results on robust stability to performance analysis results for a class of uncertain quantum systems. This class of systems involves a nominal linear quantum system and is subject to quadratic perturbations in the system Hamiltonian. Based on these two methods, coherent guaranteed cost controllers are designed for a given quantum system to achieve improved control performance. An illustrative example also shows that the quantum Popov approach can obtain less conservative results than the quantum small gain approach for the same uncertain quantum system.