Performance Bounds for Quantum Feedback Control

The limits of quantum feedback control have immediate consequences for quantum information science at large, yet remain largely unexplored. Here, we combine quantum filtering theory and moment-sum-of-squares techniques to construct a hierarchy of convex optimization problems that furnish monotonically improving, computable bounds on the best attainable performance for a broad class of quantum feedback control problems. These bounds may serve as witnesses of fundamental limitations, optimality certificates, or performance targets. We prove convergence of the bounds to the optimal control performance under technical conditions and demonstrate the practical utility of our approach by designing certifiably near-optimal controllers for a qubit in a cavity subjected to photon counting and homodyne detection measurements.