Stochastic optimal control of open quantum systems

Abstract: We address the generic problem of optimal quantum state preparation for open quantum systems. It is well known that open quantum systems can be simulated by quantum trajectories described by a stochastic Schr\"odinger equation. In this context, the state preparation becomes a stochastic optimal control (SOC) problem. The latter requires the solution of the Hamilton-Jacobi-Bellman equation, which is, in general, challenging to solve. A notable exception are the so-called path integral (PI) control problems, for which one can estimate the optimal control solution by direct sampling of the cost objective. In this work, we derive a class of quantum state preparation problems that are amenable to PI control techniques, and propose a corresponding algorithm, which we call Quantum Diffusion Control (QDC). Unlike conventional quantum control algorithms, QDC avoids computing gradients of the cost function to determine the optimal control. Instead, it employs adaptive importance sampling, a technique where the controls are iteratively improved based on global averages over quantum trajectories. We also demonstrate that QDC, used as an annealer in the environmental coupling strength, finds high accuracy solutions for unitary (noiseless) quantum control problems. We further discuss the implementation of this technique on quantum hardware. We illustrate the effectiveness of our approach through examples of open-loop control for single- and multi-qubit systems.