Path integral control of open quantum systems

We investigate open-loop quantum state preparation for a class of open quantum systems whose dynamics follow a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation that admits a trajectory-based stochastic representation. The deterministic control objective is reformulated as a stochastic optimal control problem -- interpreting stochasticity as a methodological tool akin to stochastic Schr\"odinger equation unravelings -- which situates the problem within the path integral control framework. For the class of GKLS generators under consideration, this reformulation leads to an explicit expression for the optimal control as a weighted average over stochastic quantum trajectories, thereby eliminating the need for gradient evaluations. Building on this theoretical result, we derive a control update rule for piecewise-constant control pulses and demonstrate that adaptive importance sampling progressively enhances the control estimator during optimization, culminating in the algorithm we term Path integral Quantum Control (PiQC). We further introduce an annealed variant of PiQC, wherein a synthetic noise schedule gradually steers open-system trajectories toward closed-system dynamics, enabling high-fidelity unitary state preparation. Numerical studies on a dissipative single-qubit system and a multi-qubit Nuclear Magnetic Resonance model verify that PiQC yields precise open-loop controls and displays robustness to Hamiltonian perturbations. We propose PiQC as a trajectory-based alternative to gradient-based approaches, which might offer a viable solution in quantum control problems where gradient computation is infeasible or computationally demanding.