Sum-of-Squares Bounds for Quantum Optimal Control

Flemming Holtorf, Frank Schäfer, Julian Arnold, Christopher Rackauckas, Alan Edelman

Abstract: The precise control of devices on the quantum scale is crucial for all quantum computing applications. The limitations of quantum control are therefore of great interest to those developing quantum hardware or researching quantum information protocols. We show how moment-sum-of-squares techniques can be combined with quantum dynamics and filtering theory to construct a hierarchy of convex optimization problems that furnish hard, computable bounds on the best achievable performance in quantum control tasks. These bounds can serve as witnesses of fundamental limitations, certificates of optimality, or performance targets and thus complement existing controller design strategies. In particular, our work allows us to compute bounds for common quantum control tasks, such as maximizing the fidelity of the final quantum state relative to a target state given a finite time budget or finding the minimum time required to reach the target state with near-unit fidelity. Our work also naturally applies to feedback-controlled quantum systems subjected to continuous observations, e.g., homodyne detection or photon counting, which are of emerging interest for the next generation of quantum devices. We provide an open-source package, MarkovBounds.jl, that is built upon the optimization ecosystem in the Julia programming language and exposes a high-level interface to the proposed bounding framework.