The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal

In continuously monitored quantum systems, the feedback protocol of García-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian H_meas = r A / τ applied with gain X tilts the distribution of measurement trajectories, with X < -2 producing statistically time-reversed outcomes. Why this specific Hamiltonian achieves reversal, and how the mechanism relates to score-based diffusion models in machine learning, has remained unexplained. We compute the functional derivative of the log path probability of the quantum trajectory distribution directly in density-matrix space. Combining Girsanov's theorem applied to the measurement record, Fréchet differentiation on the Banach space of trace-class operators, and Kähler geometry on the pure-state projective manifold, we prove that δlog P_F / δρ= r A / τ= H_meas. The García-Pintos feedback Hamiltonian is the score function of the quantum trajectory distribution - exactly the object Anderson's reverse-time diffusion theorem requires for trajectory reversal. The identification extends to multi-qubit systems with independent measurement channels, where the score is a sum of local operators. Two consequences follow. First, the feedback gain X generates a continuous one-parameter family of path measures for feedback-active Hamiltonians with $[H, A] neq 0$, with X = -2 recovering the backward process in leading-order linearization - a structure absent from classical diffusion, where reversal is binary. Second, the score identification enables machine learning (ML) score estimation methods - denoising score matching, sliced score matching - to replace the analytic formula when its idealizations (unit efficiency, zero delay, Gaussian noise) fail in real experiments.