Feedback Hamiltonian Reveals Quantum Trajectory Score Function

The paper proves that the feedback Hamiltonian used in continuous monitoring, originally proposed by Garcia-Pintos, Liu, and Gorshkov, is exactly the score function of the quantum trajectory distribution. The central identity is delta log P_F / delta rho = r A / tau, which the authors equate to H_meas, the measurement feedback Hamiltonian. That equality provides a formal reason why a feedback gain X = -2 recovers the backward process in leading-order linearization, and why X < -2 produces statistically time-reversed outcomes.

The proof combines stochastic calculus and operator differentiation in trace-class spaces. Key mathematical tools include:

• •Girsanov theorem applied to the measurement record
• •Frechet differentiation on the Banach space of trace-class density operators
• •Kahler geometry on the pure-state projective manifold

These ingredients let the authors compute the functional derivative of the path log-probability directly in density-matrix space and identify the analytic score. The analysis generalizes to multi-qubit systems with independent measurement channels, where the global score decomposes into a sum of local operators. The paper also shows that feedback-active Hamiltonians with noncommuting terms generate a continuous one-parameter family of path measures, with X = -2 recovering the backward process at leading order.

This result places quantum trajectory reversal squarely within the formalism of score-based diffusion models familiar to ML practitioners. By identifying H_meas as the required score for Anderson-style reverse-time diffusion, the paper explains a previously empirical control recipe and reveals a continuous interpolation between forward and backward path measures absent in classical diffusion. Importantly, the identification allows replacing ideal analytic formulas with ML estimators, denoising score matching, sliced score matching, when real experiments deviate from unit efficiency, zero delay, or Gaussian noise assumptions.

Experimental groups should test ML-based score estimators to implement robust feedback control in nonideal settings. The multi-qubit extension invites scalable, local score learning and integration with quantum error correction and measurement-based control protocols.