Applying classical control techniques to quantum systems: entanglement versus stability margin and other limitations

Abstract

Development of robust quantum control has been challenging and there are numerous obstacles to applying classical robust control to quantum system including bilinearity, marginal stability, state preparation errors, nonlinear figures of merit. The requirement of marginal stability, while not satisfied for closed quantum systems, can be satisfied for open quantum systems where Lindbladian behavior leads to nonunitary evolution, and allows for nonzero classical stability margins, but it remains difficult to extract physical insight when classical robust control tools are applied to these systems. We consider a straightforward example of the entanglement between two qubits dissipatively coupled to a lossy cavity and analyze it using the classical stability margin and structured perturbations. We attempt, where possible, to extract physical insight from these analyses. Our aim is to highlight where classical robust control can assist in the analysis of quantum systems and identify areas where more work needs to be done to develop specific methods for quantum robust control. I. INTRODUCTIONDespite the extensive success and ongoing development of robust control theory [1], the tools developed were found not to be readily applicable to classical systems [2], and they are even less readily adaptable to quantum mechanical systems. This is increasingly problematic amidst the second quantum revolution, when quantum mechanical devices for computing, networking, sensing, and simulation are moving out of laboratories and into commercial markets [3], where robust control of these devices is critical to their utility in real-world settings [4]. These issues arise because quantum systems can be difficult to cast as linear, time-invariant control systems subject to feedback control stabilization. Rather, quantum control systems, especially those that evolve unitarily in so-called closed systems, are typically open-loop bilinear systems, although several types of feedback control for closed-loop quantum systems, using both measurementbased and coherent feedback, have been developed [4], [5].Uncertainties in quantum systems can take the form of structured perturbations in state parameters (similar to classical structured uncertainties), but these uncertainties must arise in such a way that the evolution of the system remains physical and follows the law of quantum mechanics (i.e.