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This paper derives a theoretical connection between quantum feedback control and score-based diffusion models, demonstrating that the Garcia-Pintos feedback Hamiltonian, used to reverse quantum trajectories, is equivalent to the score function of the quantum trajectory distribution. The derivation leverages Girsanov's theorem, Fr茅chet differentiation, and K盲hler geometry in density-matrix space. This equivalence allows for the application of machine learning score estimation techniques to quantum trajectory reversal, particularly when experimental conditions deviate from ideal assumptions.
Quantum trajectory reversal, previously understood through specific feedback protocols, is now shown to be fundamentally linked to score-based diffusion, opening the door to ML-driven control in noisy, real-world quantum systems.
In continuously monitored quantum systems, the feedback protocol of Garc\'ia-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian $H_{\mathrm{meas}} = r A / \tau$ 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, Fr\'echet differentiation on the Banach space of trace-class operators, and K\"ahler geometry on the pure-state projective manifold, we prove that $\delta \log P_F / \delta \rho = r A / \tau = H_{\mathrm{meas}}$. The Garc\'ia-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.
