QMaxCal: Path-Space Regularization for Open Quantum Control via Girsanov’s Theorem

Quantum optimal control is often limited by decoherence—the irreversible interaction between a system and its environment. Traditional control methods focus on pulse shape or fluence, but these do not explicitly account for how a control protocol routes a system through Hilbert space to minimize exposure to noise. The authors ask: can we derive a differentiable, path-space regularizer that directly penalizes the observable effects of decoherence on quantum trajectories?

Using the stochastic Schrödinger equation (SSE), the authors model open quantum systems as Itô diffusions. They observe that measurement records of these systems share the same noise structure but differ in their drift, which is determined by the control protocol. By applying Girsanov’s theorem, they derive a closed-form estimator for the KL divergence between trajectory distributions. They instantiate this with two regularizers:

1. Wiener KL (KL_W): Drives the system toward states where decoherence effects are minimal (e.g., the kernel of Lindblad operators).
2. Drift-variance (R_DV): Stabilizes the drift to a constant value, which is effective for systems with decoherence-free subspaces that have non-zero drift.

QMaxCal outperforms standard gradient-based and reinforcement learning baselines across five benchmarks, including a six-qubit chain calibrated to the IBM Kingston processor. The regularizers reduce infidelity by up to 50% in multi-qubit benchmarks and show significant robustness to noise model mismatch, with fidelity gains increasing as the assumed noise deviates from the true environment. Unlike fluence penalties, these regularizers specifically steer trajectories away from high-decoherence regions of state space.