Inverse Design with Fourier Neural Operators for Quantum System Control

Effective quantum control is essential for scalable quantum technologies, but existing methods face challenges in high-dimensional systems due to the high cost and non-differentiability of conventional solvers. For example, molecular systems, which host many accessible states and serve as key platforms for precision measurements, remain difficult to control efficiently. To address this, we present a Fourier Neural Operator (FNO)-based framework that learns to simulate population dynamics of hyperfine states in $Ca H^{+}$ from reference simulations. For the 6- and 12-dimensional systems considered, the FNO predicts complete trajectories accurately, given a constant driving frequency and initial state populations, while achieving speedups around 600,000× and 42,000,000×, respectively. The speedups are expected to grow further with higher system dimensionality, thereby opening the path to efficient simulations of previously intractable high-dimensional quantum systems. Leveraging the FNO’s speed and differentiability, we successfully demonstrate two inverse-design strategies for controlling the driving frequency to steer a system toward a desired final state while minimizing evolution time: (i) extensive parameter space sampling across dense parameter grids, yielding accurate solutions, and (ii) gradient-based optimization, which demands more tuning but holds greater promise due to its substantially higher scalability. These results show that neural operators can provide both accurate forward simulation and scalable inverse design, opening new avenues for quantum control in complex systems.