Convex Optimization for Quantum Control in Julia

Feedback control policies for quantum systems often lack performance targets and certificates of optimality. Here, we will show how bounds on the best possible control performance are readily computable for a wide range of quantum control problems by means of convex optimization using Julia's optimization ecosystem. We discuss how these bounds provide targets and certificates to improve the design of quantum feedback controllers.

Optimal feedback control of quantum systems plays an important role in fields such as quantum information processing and quantum sensing. Despite its relevance, however, all but the simplest quantum control problems have no known analytical solutions and even rigorous numerical approximations are usually unavailable. This can be attributed to the complex dynamics associated with quantum systems subjected to continuous observation, such as in photon counting or homodyne detection setups; these systems are described by nonlinear jump-diffusion processes.

As a consequence, the use of heuristics and approximations, often based on reinforcement learning or expert intuition, is common practice for the design of quantum control policies. While these heuristics often perform remarkably well in practice, they seldom possess a mechanism to evaluate the degree of suboptimality they introduce, leaving it purely to intuition when to terminate the controller design process. We present a convex (sum-of-squares) programming-based framework for the computation of informative bounds on the best possible control performance and show how Julia's optimization ecosystem enables its implementation. We demonstrate the utility of the approach by constructing certifiably near-optimal control policies for a continuously monitored qubit.